# Publications

### 2021

1. N. Demo, G. Ortali, G. Gustin, G. Rozza, and G. Lavini, “An efficient computational framework for naval shape design and optimization problems by means of data-driven reduced order modeling techniques”, , 14(1), pp. 211-230, 2021.
This contribution describes the implementation of a data-driven shape optimization pipeline in a naval architecture application. We adopt reduced order models in order to improve the efficiency of the overall optimization, keeping a modular and equation-free nature to target the industrial demand. We applied the above mentioned pipeline to a realistic cruise ship in order to reduce the total drag. We begin by defining the design space, generated by deforming an initial shape in a parametric way using free form deformation. The evaluation of the performance of each new hull is determined by simulating the flux via finite volume discretization of a two-phase (water and air) fluid. Since the fluid dynamics model can result very expensive—especially dealing with complex industrial geometries—we propose also a dynamic mode decomposition enhancement to reduce the computational cost of a single numerical simulation. The real-time computation is finally achieved by means of proper orthogonal decomposition with Gaussian process regression technique. Thanks to the quick approximation, a genetic optimization algorithm becomes feasible to converge towards the optimal shape.

@ARTICLE{Demo2021211,
author={Demo, N. and Ortali, G. and Gustin, G. and Rozza, G. and Lavini, G.},
title={An efficient computational framework for naval shape design and optimization problems by means of data-driven reduced order modeling techniques},
year={2021},
volume={14},
number={1},
pages={211-230},
doi={10.1007/s40574-020-00263-4},
preprint={https://www.scopus.com/inward/record.uri?eid=2-s2.0-85095429770&doi=10.1007%2fs40574-020-00263-4&partnerID=40&md5=3516e38aa8ffd8e386ecf4d472c48197},
abstract={This contribution describes the implementation of a data-driven shape optimization pipeline in a naval architecture application. We adopt reduced order models in order to improve the efficiency of the overall optimization, keeping a modular and equation-free nature to target the industrial demand. We applied the above mentioned pipeline to a realistic cruise ship in order to reduce the total drag. We begin by defining the design space, generated by deforming an initial shape in a parametric way using free form deformation. The evaluation of the performance of each new hull is determined by simulating the flux via finite volume discretization of a two-phase (water and air) fluid. Since the fluid dynamics model can result very expensive—especially dealing with complex industrial geometries—we propose also a dynamic mode decomposition enhancement to reduce the computational cost of a single numerical simulation. The real-time computation is finally achieved by means of proper orthogonal decomposition with Gaussian process regression technique. Thanks to the quick approximation, a genetic optimization algorithm becomes feasible to converge towards the optimal shape.},
}

### 2020

1. S. Ali, F. Ballarin, and G. Rozza, “Stabilized reduced basis methods for parametrized steady Stokes and Navier-Stokes equations”, Computers & Mathematics with Applications, 2020.
It is well known in the Reduced Basis approximation of saddle point problems that the Galerkin projection on the reduced space does not guarantee the inf-sup approximation stability even if a stable high fidelity method was used to generate snapshots. For problems in computational fluid dynamics, the lack of inf-sup stability is reflected by the inability to accurately approximate the pressure field. In this context, inf-sup stability is usually recovered through the enrichment of the velocity space with suitable supremizer functions. The main goal of this work is to propose an alternative approach, which relies on the residual based stabilization techniques customarily employed in the Finite Element literature, such as Brezzi-Pitkaranta, Franca-Hughes, streamline upwind Petrov-Galerkin, Galerkin Least Square. In the spirit of offline-online reduced basis computational splitting, two such options are proposed, namely offline-only stabilization and offline-online stabilization. These approaches are then compared to (and combined with) the state of the art supremizer enrichment approach. Numerical results are discussed, highlighting that the proposed methodology allows to obtain smaller reduced basis spaces (i.e., neglecting supremizer enrichment) for which a modified inf-sup stability is still preserved at the reduced order level.

@article{AliBallarinRozza2020,
author = {Shafqat Ali and Francesco Ballarin and Gianluigi Rozza},
title = {Stabilized reduced basis methods for parametrized steady Stokes and Navier-Stokes equations},
year = {2020},
preprint = {https://arxiv.org/abs/2001.00820},
journal = {Computers & Mathematics with Applications},
doi = {10.1016/j.camwa.2020.03.019},
abstract = {It is well known in the Reduced Basis approximation of saddle point problems that the Galerkin projection on the reduced space does not guarantee the inf-sup approximation stability even if a stable high fidelity method was used to generate snapshots. For problems in computational fluid dynamics, the lack of inf-sup stability is reflected by the inability to accurately approximate the pressure field. In this context, inf-sup stability is usually recovered through the enrichment of the velocity space with suitable supremizer functions. The main goal of this work is to propose an alternative approach, which relies on the residual based stabilization techniques customarily employed in the Finite Element literature, such as Brezzi-Pitkaranta, Franca-Hughes, streamline upwind Petrov-Galerkin, Galerkin Least Square. In the spirit of offline-online reduced basis computational splitting, two such options are proposed, namely offline-only stabilization and offline-online stabilization. These approaches are then compared to (and combined with) the state of the art supremizer enrichment approach. Numerical results are discussed, highlighting that the proposed methodology allows to obtain smaller reduced basis spaces (i.e., neglecting supremizer enrichment) for which a modified inf-sup stability is still preserved at the reduced order level.}
}

2. F. Ballarin, T. Chacón Rebollo, E. Delgado Ávila, M. Gómez Mármol, and G. Rozza, “Certified Reduced Basis VMS-Smagorinsky model for natural convection flow in a cavity with variable height”, Computers & Mathematics with Applications, 80(5), pp. 973-989, 2020.
In this work we present a Reduced Basis VMS-Smagorinsky Boussinesq model, applied to natural convection problems in a variable height cavity, in which the buoyancy forces are involved. We take into account in this problem both physical and geometrical parametrizations, considering the Rayleigh number as a parameter, so as the height of the cavity. We perform an Empirical Interpolation Method to approximate the sub-grid eddy viscosity term that lets us obtain an affine decomposition with respect to the parameters. We construct an a posteriori error estimator, based upon the Brezzi–Rappaz–Raviart theory, used in the greedy algorithm for the selection of the basis functions. Finally we present several numerical tests for different parameter configuration.

@article{BallarinChaconDelgadoGomezRozza2020,
author = {Ballarin, Francesco and Chacón Rebollo, Tomás and Delgado Ávila, Enrique and Gómez Mármol, Macarena and Rozza, Gianluigi},
title = {Certified Reduced Basis VMS-Smagorinsky model for natural convection flow in a cavity with variable height},
journal = {Computers & Mathematics with Applications},
volume = {80},
number = {5},
pages = {973-989},
year = {2020},
preprint = {https://arxiv.org/abs/1902.05729},
doi = {10.1016/j.camwa.2020.05.013},
abstract = {In this work we present a Reduced Basis VMS-Smagorinsky Boussinesq model, applied to natural convection problems in a variable height cavity, in which the buoyancy forces are involved. We take into account in this problem both physical and geometrical parametrizations, considering the Rayleigh number as a parameter, so as the height of the cavity. We perform an Empirical Interpolation Method to approximate the sub-grid eddy viscosity term that lets us obtain an affine decomposition with respect to the parameters. We construct an a posteriori error estimator, based upon the Brezzi–Rappaz–Raviart theory, used in the greedy algorithm for the selection of the basis functions. Finally we present several numerical tests for different parameter configuration.}
}

3. N. Demo, M. Tezzele, and G. Rozza, “A supervised learning approach involving active subspaces for an efficient genetic algorithm in high-dimensional optimization problems”, 2020.
In this work, we present an extension of the genetic algorithm (GA) which exploits the active subspaces (AS) property to evolve the individuals on a lower dimensional space. In many cases, GA requires in fact more function evaluations than others optimization method to converge to the optimum. Thus, complex and high-dimensional functions may result intractable with the standard algorithm. To address this issue, we propose to linearly map the input parameter space of the original function onto its AS before the evolution, performing the mutation and mate processes in a lower dimensional space. In this contribution, we describe the novel method called ASGA, presenting differences and similarities with the standard GA method. We test the proposed method over n-dimensional benchmark functions – Rosenbrock, Ackley, Bohachevsky, Rastrigin, Schaffer N. 7, and Zakharov – and finally we apply it to an aeronautical shape optimization problem.

@unpublished{DemoTezzeleRozza2020,
author = {Nicola Demo and Marco Tezzele and Gianluigi Rozza},
title = {A supervised learning approach involving active subspaces for an efficient genetic algorithm in high-dimensional optimization problems},
year = {2020},
abstract = {In this work, we present an extension of the genetic algorithm (GA) which exploits the active subspaces (AS) property to evolve the individuals on a lower dimensional space. In many cases, GA requires in fact more function evaluations than others optimization method to converge to the optimum. Thus, complex and high-dimensional functions may result intractable with the standard algorithm. To address this issue, we propose to linearly map the input parameter space of the original function onto its AS before the evolution, performing the mutation and mate processes in a lower dimensional space. In this contribution, we describe the novel method called ASGA, presenting differences and similarities with the standard GA method. We test the proposed method over n-dimensional benchmark functions -- Rosenbrock, Ackley, Bohachevsky, Rastrigin, Schaffer N. 7, and Zakharov -- and finally we apply it to an aeronautical shape optimization problem.},
preprint = {https://arxiv.org/abs/2006.07282},
}

4. M. Gadalla, M. Cianferra, M. Tezzele, G. Stabile, A. Mola, and G. Rozza, “On the comparison of LES data-driven reduced order approaches for hydroacoustic analysis”, 2020.
In this work, Dynamic Mode Decomposition (DMD) and Proper Orthogonal Decomposition (POD) methodologies are applied to hydroacoustic dataset computed using Large Eddy Simulation (LES) coupled with Ffowcs Williams and Hawkings (FWH) analogy. First, a low-dimensional description of the flow fields is presented with modal decomposition analysis. Sensitivity towards the DMD and POD bases truncation rank is discussed, and extensive dataset is provided to demonstrate the ability of both algorithms to reconstruct the flow fields with all the spatial and temporal frequencies necessary to support accurate noise evaluation. Results show that while DMD is capable to capture finer coherent structures in the wake region for the same amount of employed modes, reconstructed flow fields using POD exhibit smaller magnitudes of global spatiotemporal errors compared with DMD counterparts. Second, a separate set of DMD and POD modes generated using half the snapshots is employed into two data-driven reduced models respectively, based on DMD mid cast and POD with Interpolation (PODI). In that regard, results confirm that the predictive character of both reduced approaches on the flow fields is sufficiently accurate, with a relative superiority of PODI results over DMD ones. This infers that, discrepancies induced due to interpolation errors in PODI is relatively low compared with errors induced by integration and linear regression operations in DMD, for the present setup. Finally, a post processing analysis on the evaluation of FWH acoustic signals utilizing reduced fluid dynamic fields as input demonstrates that both DMD and PODI data-driven reduced models are efficient and sufficiently accurate in predicting acoustic noises.

@unpublished{GadallaCianferraTezzeleStabileMolaRozza2020,
author = {Mahmoud Gadalla and Marta Cianferra and Marco Tezzele and Giovanni Stabile and Andrea Mola and Gianluigi Rozza},
title = {On the comparison of LES data-driven reduced order approaches for hydroacoustic analysis},
year = {2020},
preprint = {https://arxiv.org/abs/2006.14428},
abstract = {In this work, Dynamic Mode Decomposition (DMD) and Proper Orthogonal Decomposition (POD) methodologies are applied to hydroacoustic dataset computed using Large Eddy Simulation (LES) coupled with Ffowcs Williams and Hawkings (FWH) analogy. First, a low-dimensional description of the flow fields is presented with modal decomposition analysis. Sensitivity towards the DMD and POD bases truncation rank is discussed, and extensive dataset is provided to demonstrate the ability of both algorithms to reconstruct the flow fields with all the spatial and temporal frequencies necessary to support accurate noise evaluation. Results show that while DMD is capable to capture finer coherent structures in the wake region for the same amount of employed modes, reconstructed flow fields using POD exhibit smaller magnitudes of global spatiotemporal errors compared with DMD counterparts. Second, a separate set of DMD and POD modes generated using half the snapshots is employed into two data-driven reduced models respectively, based on DMD mid cast and POD with Interpolation (PODI). In that regard, results confirm that the predictive character of both reduced approaches on the flow fields is sufficiently accurate, with a relative superiority of PODI results over DMD ones. This infers that, discrepancies induced due to interpolation errors in PODI is relatively low compared with errors induced by integration and linear regression operations in DMD, for the present setup. Finally, a post processing analysis on the evaluation of FWH acoustic signals utilizing reduced fluid dynamic fields as input demonstrates that both DMD and PODI data-driven reduced models are efficient and sufficiently accurate in predicting acoustic noises.}
}

5. S. Georgaka, G. Stabile, K. Star, G. Rozza, and M. J. Bluck, “A hybrid reduced order method for modelling turbulent heat transfer problems”, Computers & Fluids, 208, pp. 104615, 2020.
A parametric, hybrid reduced order model approach based on the Proper Orthogonal Decomposition with both Galerkin projection and interpolation based on Radial Basis Functions method is presented. This method is tested against a case of turbulent non-isothermal mixing in a T-junction pipe, a common ow arrangement found in nuclear reactor cooling systems. The reduced order model is derived from the 3D unsteady, incompressible Navier-Stokes equations weakly coupled with the energy equation. For high Reynolds numbers, the eddy viscosity and eddy diffusivity are incorporated into the reduced order model with a Proper Orthogonal Decomposition (nested and standard) with Interpolation (PODI), where the interpolation is performed using Radial Basis Functions. The reduced order solver, obtained using a k-{\omega} SST URANS full order model, is tested against the full order solver in a 3D T-junction pipe with parametric velocity inlet boundary conditions.

@article{GeorgakaStabileStarRozzaBluck2020,
author = {Sokratia Georgaka and Giovanni Stabile and Kelbij Star and Gianluigi Rozza and Michael J. Bluck},
journal = {Computers & Fluids},
title = {A hybrid reduced order method for modelling turbulent heat transfer problems},
year = {2020},
pages = {104615},
volume = {208},
abstract = {A parametric, hybrid reduced order model approach based on the Proper Orthogonal Decomposition with both Galerkin projection and interpolation based on Radial Basis Functions method is presented. This method is tested against a case of turbulent non-isothermal mixing in a T-junction pipe, a common ow arrangement found in nuclear reactor cooling systems. The reduced order model is derived from the 3D unsteady, incompressible Navier-Stokes equations weakly coupled with the energy equation. For high Reynolds numbers, the eddy viscosity and eddy diffusivity are incorporated into the reduced order model with a Proper Orthogonal Decomposition (nested and standard) with Interpolation (PODI), where the interpolation is performed using Radial Basis Functions. The reduced order solver, obtained using a k-{\omega} SST URANS full order model, is tested against the full order solver in a 3D T-junction pipe with parametric velocity inlet boundary conditions.},
doi = {10.1016/j.compfluid.2020.104615},
preprint = {https://arxiv.org/abs/1906.08725},
}

6. M. Girfoglio, A. Quaini, and G. Rozza, “A POD-Galerkin reduced order model for a LES filtering approach”, 2020.
We propose a Proper Orthogonal Decomposition (POD)-Galerkin based Reduced Order Model (ROM) for a Leray model. For the implementation of the model, we combine a two-step algorithm called Evolve-Filter (EF) with a computationally efficient finite volume method. The main novelty of the proposed approach relies in applying spatial filtering both for the collection of the snapshots and in the reduced order model, as well as in considering the pressure field at reduced level. In both steps of the EF algorithm, velocity and pressure fields are approximated by using different POD basis and coefficients. For the reconstruction of the pressures fields, we use a pressure Poisson equation approach. We test our ROM on two benchmark problems: 2D and 3D unsteady flow past a cylinder at Reynolds number 0 <= Re <= 100. The accuracy of the reduced order model is assessed against results obtained with the full order model. For the 2D case, a parametric study with respect to the filtering radius is also presented.

@unpublished{GirfoglioQuainiRozza2020,
author = {Michele Girfoglio and Annalisa Quaini and Gianluigi Rozza},
title = {A POD-Galerkin reduced order model for a LES filtering approach},
year = {2020},
preprint = {https://arxiv.org/abs/2009.13593},
abstract = {We propose a Proper Orthogonal Decomposition (POD)-Galerkin based Reduced Order Model (ROM) for a Leray model. For the implementation of the model, we combine a two-step algorithm called Evolve-Filter (EF) with a computationally efficient finite volume method. The main novelty of the proposed approach relies in applying spatial filtering both for the collection of the snapshots and in the reduced order model, as well as in considering the pressure field at reduced level. In both steps of the EF algorithm, velocity and pressure fields are approximated by using different POD basis and coefficients. For the reconstruction of the pressures fields, we use a pressure Poisson equation approach. We test our ROM on two benchmark problems: 2D and 3D unsteady flow past a cylinder at Reynolds number 0 <= Re <= 100. The accuracy of the reduced order model is assessed against results obtained with the full order model. For the 2D case, a parametric study with respect to the filtering radius is also presented.}
}

7. M. Girfoglio, F. Ballarin, G. Infantino, F. Nicolò, A. Montalto, G. Rozza, R. Scrofani, M. Comisso, and F. Musumeci, "Non-intrusive PODI-ROM for patient-specific aortic blood flow in presence of a LVAD device", 2020.
Left ventricular assist devices (LVADs) are used to provide haemodynamic support to patients with critical cardiac failure. Severe complications can occur because of the modifications of the blood flow in the aortic region. In this work, the effect of a continuous flow LVAD device on the aortic flow is investigated by means of a non-intrusive reduced order model (ROM) built using the proper orthogonal decomposition with interpolation (PODI) method. The full order model (FOM) is represented by the incompressible Navier-Stokes equations discretized by using a Finite Volume (FV) technique, coupled with three-element Windkessel models to enforce outlet boundary conditions in a multi-scale approach. A patient-specific framework is proposed: a personalized geometry reconstructed from Computed Tomography (CT) images is used and the individualization of the coefficients of the three-element Windkessel models is based on experimental data provided by the Right Heart Catheterization (RCH) and Echocardiography (ECHO) tests. Pre-surgery configuration is also considered at FOM level in order to further validate the model. A parametric study with respect to the LVAD flow rate is considered. The accuracy of the reduced order model is assessed against results obtained with the full order model.

@unpublished{GirfoglioBallarinInfantinoNicoloMontaltoRozzaScrofaniComissoMusumeci2020,
author = {Michele Girfoglio and Francesco Ballarin and Giuseppe Infantino and Francesca Nicolò and Andrea Montalto and Gianluigi Rozza and Roberto Scrofani and Marina Comisso and Francesco Musumeci},
title = {Non-intrusive PODI-ROM for patient-specific aortic blood flow in presence of a LVAD device},
year = {2020},
preprint = {https://arxiv.org/abs/2007.03527},
abstract = {Left ventricular assist devices (LVADs) are used to provide haemodynamic support to patients with critical cardiac failure. Severe complications can occur because of the modifications of the blood flow in the aortic region. In this work, the effect of a continuous flow LVAD device on the aortic flow is investigated by means of a non-intrusive reduced order model (ROM) built using the proper orthogonal decomposition with interpolation (PODI) method. The full order model (FOM) is represented by the incompressible Navier-Stokes equations discretized by using a Finite Volume (FV) technique, coupled with three-element Windkessel models to enforce outlet boundary conditions in a multi-scale approach. A patient-specific framework is proposed: a personalized geometry reconstructed from Computed Tomography (CT) images is used and the individualization of the coefficients of the three-element Windkessel models is based on experimental data provided by the Right Heart Catheterization (RCH) and Echocardiography (ECHO) tests. Pre-surgery configuration is also considered at FOM level in order to further validate the model. A parametric study with respect to the LVAD flow rate is considered. The accuracy of the reduced order model is assessed against results obtained with the full order model.}
}

8. N. Giuliani, M. W. Hess, A. DeSimone, and G. Rozza, "MicroROM: An Efficient and Accurate Reduced Order Method to Solve Many-Query Problems in Micro-Motility", 2020.
In the study of micro-swimmers, both artificial and biological ones, many-query problems arise naturally. Even with the use of advanced high performance computing (HPC), it is not possible to solve this kind of problems in an acceptable amount of time. Various approximations of the Stokes equation have been considered in the past to ease such computational efforts but they introduce non-negligible errors that can easily make the solution of the problem inaccurate and unreliable. Reduced order modeling solves this issue by taking advantage of a proper subdivision between a computationally expensive offline phase and a fast and efficient online stage. This work presents the coupling of Boundary Element Method (BEM) and Reduced Basis (RB) Reduced Order Modeling (ROM) in two models of practical interest, obtaining accurate and reliable solutions to different many-query problems. Comparisons of standard reduced order modeling approaches in different simulation settings and a comparison to typical approximations to Stokes equations are also shown. Different couplings between a solver based on a HPC boundary element method for micro-motility problems and reduced order models are presented in detail. The methodology is tested on two different models: a robotic-bacterium-like and an Eukaryotic-like swimmer, and in each case two resolution strategies for the swimming problem, the split and monolithic one, are used as starting points for the ROM. An efficient and accurate reconstruction of the performance of interest is achieved in both cases proving the effectiveness of our strategy.

@unpublished{GiulianiHessDeSimoneRozza2020,
author = {Nicola Giuliani and Martin W. Hess and Antonio DeSimone and Gianluigi Rozza},
title = {MicroROM: An Efficient and Accurate Reduced Order Method to Solve Many-Query Problems in Micro-Motility},
year = {2020},
preprint = {https://arxiv.org/abs/2006.13836},
abstract = {In the study of micro-swimmers, both artificial and biological ones, many-query problems arise naturally. Even with the use of advanced high performance computing (HPC), it is not possible to solve this kind of problems in an acceptable amount of time. Various approximations of the Stokes equation have been considered in the past to ease such computational efforts but they introduce non-negligible errors that can easily make the solution of the problem inaccurate and unreliable. Reduced order modeling solves this issue by taking advantage of a proper subdivision between a computationally expensive offline phase and a fast and efficient online stage.
This work presents the coupling of Boundary Element Method (BEM) and Reduced Basis (RB) Reduced
Order Modeling (ROM) in two models of practical interest, obtaining accurate and reliable solutions to different many-query problems. Comparisons of standard reduced order modeling approaches in different simulation settings and a comparison to typical approximations to Stokes equations are also shown. Different couplings between a solver based on a HPC boundary element method for micro-motility problems and reduced order models are presented in detail. The methodology is tested on two different models: a robotic-bacterium-like and an Eukaryotic-like swimmer, and in each case two resolution strategies for the swimming problem, the split and monolithic one, are used as starting points for the ROM. An efficient and accurate reconstruction of the performance of interest is achieved in both cases proving the effectiveness of our strategy. }
}

9. M. W. Hess, A. Quaini, and G. Rozza, "A comparison of reduced-order modeling approaches for PDEs with bifurcating solutions", 2020.
This paper focuses on reduced-order models (ROMs) built for the efficient treatment of PDEs having solutions that bifurcate as the values of multiple input parameters change. First, we consider a method called local ROM that uses k-means algorithm to cluster snapshots and construct local POD bases, one for each cluster. We investigate one key ingredient of this approach: the local basis selection criterion. Several criteria are compared and it is found that a criterion based on a regression artificial neural network (ANN) provides the most accurate results for a channel flow problem exhibiting a supercritical pitchfork bifurcation. The same benchmark test is then used to compare the local ROM approach with the regression ANN selection criterion to an established global projection-based ROM and a recently proposed ANN based method called POD-NN. We show that our local ROM approach gains more than an order of magnitude in accuracy over the global projection-based ROM. However, the POD-NN provides consistently more accurate approximations than the local projection-based ROM.

@unpublished{HessQuainiRozza2020,
author = {Martin W. Hess and Annalisa Quaini and Gianluigi Rozza},
title = {A comparison of reduced-order modeling approaches for PDEs with bifurcating solutions},
year = {2020},
preprint = {https://arxiv.org/abs/2010.07370},
abstract = {This paper focuses on reduced-order models (ROMs) built for the efficient treatment of PDEs having solutions that bifurcate as the values of multiple input parameters change. First, we consider a method called local ROM that uses k-means algorithm to cluster snapshots and construct local POD bases, one for each cluster. We investigate one key ingredient of this approach: the local basis selection criterion. Several criteria are compared and it is found that a criterion based on a regression artificial neural network (ANN) provides the most accurate results for a channel flow problem exhibiting a supercritical pitchfork bifurcation. The same benchmark test is then used to compare the local ROM approach with the regression ANN selection criterion to an established global projection-based ROM and a recently proposed ANN based method called POD-NN. We show that our local ROM approach gains more than an order of magnitude in accuracy over the global projection-based ROM. However, the POD-NN provides consistently more accurate approximations than the local projection-based ROM.}
}

10. M. Hess, A. Quaini, and G. Rozza, "Reduced Basis Model Order Reduction for Navier-Stokes equations in domains with walls of varying curvature", International Journal of Computational Fluid Dynamics, 34(2), pp. 119-126, 2020.
We consider the Navier-Stokes equations in a channel with a narrowing and walls of varying curvature. By applying the empirical interpolation method to generate an affine parameter dependency, the offline-online procedure can be used to compute reduced order solutions for parameter variations. The reduced order space is computed from the steady-state snapshot solutions by a standard POD procedure. The model is discretised with high-order spectral element ansatz functions, resulting in 4752 degrees of freedom. The proposed reduced order model produces accurate approximations of steady-state solutions for a wide range of geometries and kinematic viscosity values. The application that motivated the present study is the onset of asymmetries (i.e., symmetry breaking bifurcation) in blood flow through a regurgitant mitral valve, depending on the Reynolds number and the valve shape. Through our computational study, we found that the critical Reynolds number for the symmetry breaking increases as the wall curvature increases.

@article{HessQuainiRozza2020,
author = {Hess, Martin and Quaini, Annalisa and Rozza, Gianluigi},
title = {Reduced Basis Model Order Reduction for Navier-Stokes equations in domains with walls of varying curvature},
journal = {International Journal of Computational Fluid Dynamics},
volume = {34},
number = {2},
pages = {119-126},
year = {2020},
doi = {10.1080/10618562.2019.1645328},
preprint = {https://arxiv.org/abs/1901.03708},
abstract = {We consider the Navier-Stokes equations in a channel with a narrowing and walls of varying curvature. By applying the empirical interpolation method to generate an affine parameter dependency, the offline-online procedure can be used to compute reduced order solutions for parameter variations. The reduced order space is computed from the steady-state snapshot solutions by a standard POD procedure. The model is discretised with high-order spectral element ansatz functions, resulting in 4752 degrees of freedom. The proposed reduced order model produces accurate approximations of steady-state solutions for a wide range of geometries and kinematic viscosity values. The application that motivated the present study is the onset of asymmetries (i.e., symmetry breaking bifurcation) in blood flow through a regurgitant mitral valve, depending on the Reynolds number and the valve shape. Through our computational study, we found that the critical Reynolds number for the symmetry breaking increases as the wall curvature increases.}
}

11. M. W. Hess, A. Quaini, and G. Rozza, "A Spectral Element Reduced Basis Method for Navier–Stokes Equations with Geometric Variations", in Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018, 2020, pp. 561–571.
We consider the Navier-Stokes equations in a channel with a narrowing of varying height. The model is discretized with high-order spectral element ansatz functions, resulting in 6372 degrees of freedom. The steady-state snapshot solutions define a reduced order space through a standard POD procedure. The reduced order space allows to accurately and efficiently evaluate the steady-state solutions for different geometries. In particular, we detail different aspects of implementing the reduced order model in combination with a spectral element discretization. It is shown that an expansion in element-wise local degrees of freedom can be combined with a reduced order modelling approach to enhance computational times in parametric many-query scenarios.

@InProceedings{HessQuainiRozza2020,
author = {Hess, Martin W. and Quaini, Annalisa and Rozza, Gianluigi},
editor = {Sherwin, Spencer J. and Moxey, David and Peir{\'o}, Joaquim and Vincent, Peter E. and Schwab, Christoph},
title = {A Spectral Element Reduced Basis Method for Navier--Stokes Equations with Geometric Variations},
booktitle = {Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018},
year = {2020},
publisher = {Springer International Publishing},
pages = {561--571},
preprint = {https://arxiv.org/abs/1812.11051},
doi = {10.1007/978-3-030-39647-3_45},
abstract = {We consider the Navier-Stokes equations in a channel with a narrowing of varying height. The model is discretized with high-order spectral element ansatz functions, resulting in 6372 degrees of freedom. The steady-state snapshot solutions define a reduced order space through a standard POD procedure. The reduced order space allows to accurately and efficiently evaluate the steady-state solutions for different geometries. In particular, we detail different aspects of implementing the reduced order model in combination with a spectral element discretization. It is shown that an expansion in element-wise local degrees of freedom can be combined with a reduced order modelling approach to enhance computational times in parametric many-query scenarios.},
}

12. S. Hijazi, G. Stabile, A. Mola, and G. Rozza, "Data-driven POD-Galerkin reduced order model for turbulent flows", Journal of Computational Physics, 416, pp. 109513, 2020.
In this work we present a Reduced Order Model which is specifically designed to deal with turbulent flows in a finite volume setting. The method used to build the reduced order model is based on the idea of merging/combining projection-based techniques with data-driven reduction strategies. In particular, the work presents a mixed strategy that exploits a data-driven reduction method to approximate the eddy viscosity solution manifold and a classical POD-Galerkin projection approach for the velocity and the pressure fields, respectively. The newly proposed reduced order model has been validated on benchmark test cases in both steady and unsteady settings with Reynolds up to $Re=O(10^5)$.

@article{HijaziStabileMolaRozza2020,
author = {Saddam Hijazi and Giovanni Stabile and Andrea Mola and Gianluigi Rozza},
title = {Data-driven POD-Galerkin reduced order model for turbulent flows},
year = {2020},
preprint = {https://arxiv.org/abs/1907.09909},
abstract = {In this work we present a Reduced Order Model which is specifically designed to deal with turbulent flows in a finite volume setting. The method used to build the reduced order model is based on the idea of merging/combining projection-based techniques with data-driven reduction strategies. In particular, the work presents a mixed strategy that exploits a data-driven reduction method to approximate the eddy viscosity solution manifold and a classical POD-Galerkin projection approach for the velocity and the pressure fields, respectively. The newly proposed reduced order model has been validated on benchmark test cases in both steady and unsteady settings with Reynolds up to $Re=O(10^5)$.},
journal = {Journal of Computational Physics},
doi = {10.1016/j.jcp.2020.109513},
volume = {416},
pages = {109513},
}

13. S. Hijazi, G. Stabile, A. Mola, and G. Rozza, "Non-Intrusive Polynomial Chaos Method Applied to Problems in Computational Fluid Dynamics with a Comparison to Proper Orthogonal Decomposition", in QUIET Selected Contributions, H. van Brummelen, A. Corsini, S. Perotto, and G. Rozza (eds.), Springer International Publishing, 2020.
In this work, Uncertainty Quantification (UQ) based on non-intrusive Polynomial Chaos Expansion (PCE) is applied to the CFD problem of the flow past an airfoil with parameterized angle of attack and inflow velocity. To limit the computational cost associated with each of the simulations required by the non-intrusive UQ algorithm used, we resort to a Reduced Order Model (ROM) based on Proper Orthogonal Decomposition (POD)-Galerkin approach. A first set of results is presented to characterize the accuracy of the POD-Galerkin ROM developed approach with respect to the Full Order Model (FOM) solver (OpenFOAM). A further analysis is then presented to assess how the UQ results are affected by substituting the FOM predictions with the surrogate ROM ones.

@inbook{HijaziStabileMolaRozza2020,
author = {Saddam Hijazi and Giovanni Stabile and Andrea Mola and Gianluigi Rozza},
editor = {van Brummelen, Harald and Corsini, Alessandro and Perotto, Simona and Rozza, Gianluigi},
chapter = {Non-Intrusive Polynomial Chaos Method Applied to Problems in Computational Fluid Dynamics with a Comparison to Proper Orthogonal Decomposition},
year = {2020},
preprint = {https://arxiv.org/abs/1901.02285},
abstract = {In this work, Uncertainty Quantification (UQ) based on non-intrusive Polynomial Chaos Expansion (PCE) is applied to the CFD problem of the flow past an airfoil with parameterized angle of attack and inflow velocity. To limit the computational cost associated with each of the simulations required by the non-intrusive UQ algorithm used, we resort to a Reduced Order Model (ROM) based on Proper Orthogonal Decomposition (POD)-Galerkin approach. A first set of results is presented to characterize the accuracy of the POD-Galerkin ROM developed approach with respect to the Full Order Model (FOM) solver (OpenFOAM). A further analysis is then presented to assess how the UQ results are affected by substituting the FOM predictions with the surrogate ROM ones.},
publisher = {Springer International Publishing},
booktitle = {QUIET Selected Contributions},
}

14. S. Hijazi, S. Ali, G. Stabile, F. Ballarin, and G. Rozza, "The Effort of Increasing Reynolds Number in Projection-Based Reduced Order Methods: From Laminar to Turbulent Flows", in Numerical Methods for Flows: FEF 2017 Selected Contributions, H. van Brummelen, A. Corsini, S. Perotto, and G. Rozza (eds.), Springer International Publishing, pp. 245–264, 2020.
We present in this double contribution two different reduced order strategies for incompressible parameterized Navier-Stokes equations characterized by varying Reynolds numbers. The first strategy deals with low Reynolds number (laminar flow) and is based on a stabilized finite element method during the offline stage followed by a Galerkin projection on reduced basis spaces generated by a greedy algorithm. The second methodology is based on a full order finite volume discretization. The latter methodology will be used for flows with moderate to high Reynolds number characterized by turbulent patterns. For the treatment of the mentioned turbulent flows at the reduced order level, a new POD-Galerkin approach is proposed. The new approach takes into consideration the contribution of the eddy viscosity also during the online stage and is based on the use of interpolation. The two methodologies are tested on classic benchmark test cases.

@inbook{HijaziAliStabileBallarinRozza2020,
author = {Hijazi, Saddam and Ali, Shafqat and Stabile, Giovanni and Ballarin, Francesco and Rozza, Gianluigi},
editor = {van Brummelen, Harald and Corsini, Alessandro and Perotto, Simona and Rozza, Gianluigi},
chapter = {The Effort of Increasing Reynolds Number in Projection-Based Reduced Order Methods: From Laminar to Turbulent Flows},
booktitle = {Numerical Methods for Flows: FEF 2017 Selected Contributions},
year = {2020},
publisher = {Springer International Publishing},
pages = {245--264},
abstract = {We present in this double contribution two different reduced order strategies for incompressible parameterized Navier-Stokes equations characterized by varying Reynolds numbers. The first strategy deals with low Reynolds number (laminar flow) and is based on a stabilized finite element method during the offline stage followed by a Galerkin projection on reduced basis spaces generated by a greedy algorithm. The second methodology is based on a full order finite volume discretization. The latter methodology will be used for flows with moderate to high Reynolds number characterized by turbulent patterns. For the treatment of the mentioned turbulent flows at the reduced order level, a new POD-Galerkin approach is proposed. The new approach takes into consideration the contribution of the eddy viscosity also during the online stage and is based on the use of interpolation. The two methodologies are tested on classic benchmark test cases.},
doi = {10.1007/978-3-030-30705-9_22},
preprint = {https://arxiv.org/abs/1807.11370},
}

15. E. N. Karatzas and G. Rozza, "A Reduced Order Model for a stable embedded boundary parametrized Cahn-Hilliard phase-field system based on cut finite elements", 2020.
In the present work, we investigate for the first time with a cut finite element method, a parameterized fourth-order nonlinear geometrical PDE, namely the Cahn-Hilliard system. We manage to tackle the instability issues of such methods whenever strong nonlinearities appear and to utilize their flexibility of the fixed background geometry – and mesh – characteristic, through which, one can avoid e.g. in parametrized geometries the remeshing on the full order level, as well as, transformations to reference geometries on the reduced level. As a final goal, we manage to find an efficient global, concerning the geometrical manifold, and independent of geometrical changes, reduced-order basis. The POD-Galerkin approach exhibits its strength even with pseudo-random discontinuous initial data verified by numerical experiments.

@unpublished{KaratzasRozza2020,
author = {Efthymios N. Karatzas and Gianluigi Rozza},
title = {A Reduced Order Model for a stable embedded boundary parametrized Cahn-Hilliard phase-field system based on cut finite elements},
year = {2020},
preprint = {https://arxiv.org/abs/2009.01596},
abstract = {In the present work, we investigate for the first time with a cut finite element method, a parameterized fourth-order nonlinear geometrical PDE, namely the Cahn-Hilliard system. We manage to tackle the instability issues of such methods whenever strong nonlinearities appear and to utilize their flexibility of the fixed background geometry -- and mesh -- characteristic, through which, one can avoid e.g. in parametrized geometries the remeshing on the full order level, as well as, transformations to reference geometries on the reduced level. As a final goal, we manage to find an efficient global, concerning the geometrical manifold, and independent of geometrical changes, reduced-order basis. The POD-Galerkin approach exhibits its strength even with pseudo-random discontinuous initial data verified by numerical experiments. }
}

16. E. N. Karatzas, G. Stabile, N. Atallah, G. Scovazzi, and G. Rozza, "A Reduced Order Approach for the Embedded Shifted Boundary FEM and a Heat Exchange System on Parametrized Geometries", in IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 22–25, 2018, 2020, pp. 111–125.
A model order reduction technique is combined with an embedded boundary finite element method with a POD-Galerkin strategy. The proposed methodology is applied to parametrized heat transfer problems and we rely on a sufficiently refined shape-regular background mesh to account for parametrized geometries. In particular, the employed embedded boundary element method is the Shifted Boundary Method (SBM) recently proposed. This approach is based on the idea of shifting the location of true boundary conditions to a surrogate boundary, with the goal of avoiding cut cells near the boundary of the computational domain. This combination of methodologies has multiple advantages. In the first place, since the Shifted Boundary Method always relies on the same background mesh, there is no need to update the discretized parametric domain. Secondly, we avoid the treatment of cut cell elements, which usually need particular attention. Thirdly, since the whole background mesh is considered in the reduced basis construction, the SBM allows for a smooth transition of the reduced modes across the immersed domain boundary. The performances of the method are verified in two dimensional heat transfer numerical examples.

@InProceedings{KaratzasStabileAtallahScovazziRozza2020,
author = {Efthymios N. Karatzas and Giovanni Stabile and Nabib Atallah and Guglielmo Scovazzi and Gianluigi Rozza},
booktitle = {IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 22--25, 2018},
title = {A Reduced Order Approach for the Embedded Shifted Boundary FEM and a Heat Exchange System on Parametrized Geometries},
year = {2020},
editor = {Fehr, J{\"o}rg and Haasdonk, Bernard},
pages = {111--125},
publisher = {Springer International Publishing},
abstract = {A model order reduction technique is combined with an embedded boundary finite element method with a POD-Galerkin strategy. The proposed methodology is applied to parametrized heat transfer problems and we rely on a sufficiently refined shape-regular background mesh to account for parametrized geometries. In particular, the employed embedded boundary element method is the Shifted Boundary Method (SBM) recently proposed. This approach is based on the idea of shifting the location of true boundary conditions to a surrogate boundary, with the goal of avoiding cut cells near the boundary of the computational domain. This combination of methodologies has multiple advantages. In the first place, since the Shifted Boundary Method always relies on the same background mesh, there is no need to update the discretized parametric domain. Secondly, we avoid the treatment of cut cell elements, which usually need particular attention. Thirdly, since the whole background mesh is considered in the reduced basis construction, the SBM allows for a smooth transition of the reduced modes across the immersed domain boundary. The performances of the method are verified in two dimensional heat transfer numerical examples.},
doi = {10.1007/978-3-030-21013-7_8},
preprint = {https://arxiv.org/abs/1807.07753},
}

17. E. N. Karatzas, M. Nonino, F. Ballarin, and G. Rozza, "A Reduced Order Cut Finite Element method for geometrically parameterized steady and unsteady Navier-Stokes problems", 2020.
This work focuses on steady and unsteady Navier-Stokes equations in a reduced order modeling framework. The methodology proposed is based on a Proper Orthogonal Decomposition within a levelset geometry description and the problems of interest are discretized with an unfitted mesh Finite Element Method. We construct and investigate a unified and geometry independent reduced basis which overcomes many barriers and complications of the past, that may occur whenever geometrical morphings are taking place. By employing a geometry independent reduced basis, we are able to avoid remeshing and transformation to reference configurations, and we are able to handle complex geometries. This combination of a fixed background mesh in a fixed extended background geometry with reduced order techniques appears beneficial and advantageous in many industrial and engineering applications, which could not be resolved efficiently in the past.

@unpublished{KaratzasNoninoBallarinRozza2020,
author = {Efthymios N. Karatzas and Monica Nonino and Francesco Ballarin and Gianluigi Rozza},
title = {A Reduced Order Cut Finite Element method for geometrically parameterized steady and unsteady Navier-Stokes problems},
year = {2020},
preprint = {https://arxiv.org/abs/2010.04953},
abstract = {This work focuses on steady and unsteady Navier-Stokes equations in a reduced order modeling framework. The methodology proposed is based on a Proper Orthogonal Decomposition within a levelset geometry description and the problems of interest are discretized with an unfitted mesh Finite Element Method. We construct and investigate a unified and geometry independent reduced basis which overcomes many barriers and complications of the past, that may occur whenever geometrical morphings are taking place. By employing a geometry independent reduced basis, we are able to avoid remeshing and transformation to reference configurations, and we are able to handle complex geometries. This combination of a fixed background mesh in a fixed extended background geometry with reduced order techniques appears beneficial and advantageous in many industrial and engineering applications, which could not be resolved efficiently in the past.}
}

18. E. N. Karatzas, F. Ballarin, and G. Rozza, "Projection-based reduced order models for a cut finite element method in parametrized domains", Computers & Mathematics with Applications, 79(3), pp. 833–851, 2020.
This work presents a reduced order modelling technique built on a high fidelity embedded mesh finite element method. Such methods, and in particular the CutFEM method, are attractive in the generation of projection-based reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains. The combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems, avoiding remeshing as well as the reference domain formulation, often used in the reduced order modelling for boundary fitted finite element formulations. The resulting novel methodology is presented on linear elliptic and Stokes problems, together with several test cases to assess its capability. The role of a proper extension and transport of embedded solutions to a common background is analyzed in detail.

@article{KaratzasBallarinRozza2020,
author = {Karatzas, Efthymios N. and Ballarin, Francesco and Rozza, Gianluigi},
title = {Projection-based reduced order models for a cut finite element method in parametrized domains},
journal = {Computers & Mathematics with Applications},
volume = {79},
number = {3},
pages = {833--851},
year = {2020},
doi = {10.1016/j.camwa.2019.08.003},
preprint = {https://arxiv.org/abs/1901.03846},
abstract = {This work presents a reduced order modelling technique built on a high fidelity embedded mesh finite element method. Such methods, and in particular the CutFEM method, are attractive in the generation of projection-based reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains. The combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems, avoiding remeshing as well as the reference domain formulation, often used in the reduced order modelling for boundary fitted finite element formulations. The resulting novel methodology is presented on linear elliptic and Stokes problems, together with several test cases to assess its capability. The role of a proper extension and transport of embedded solutions to a common background is analyzed in detail.}
}

19. F. Pichi, A. Quaini, and G. Rozza, "A Reduced Order technique to study bifurcating phenomena: application to the Gross-Pitaevskii equation", , 2020.
We propose a computationally efficient framework to treat nonlinear partial differential equations having bifurcating solutions as one or more physical control parameters are varied. Our focus is on steady bifurcations. Plotting a bifurcation diagram entails computing multiple solutions of a parametrized, nonlinear problem, which can be extremely expensive in terms of computational time. In order to reduce these demanding computational costs, our approach combines a continuation technique and Newton's method with a Reduced Order Modeling (ROM) technique, suitably supplemented with a hyper-reduction method. To demonstrate the effectiveness of our ROM approach, we trace the steady solution branches of a nonlinear Schrödinger equation, called Gross-Pitaevskii equation, as one or two physical parameters are varied. In the two parameter study, we show that our approach is 60 times faster in constructing a bifurcation diagram than a standard Full Order Method.} journal = {SIAM Journal on Scientific Computing

@article{PichiQuainiRozza2020,
author = {Pichi, Federico and Quaini, Annalisa and Rozza, Gianluigi},
title = {A Reduced Order technique to study bifurcating phenomena: application to the Gross-Pitaevskii equation},
year = {2020},
preprint = {https://arxiv.org/abs/1907.07082},
abstract = {We propose a computationally efficient framework to treat nonlinear partial differential equations having bifurcating solutions as one or more physical control parameters are varied. Our focus is on steady bifurcations. Plotting a bifurcation diagram entails computing multiple solutions of a parametrized, nonlinear problem, which can be extremely expensive in terms of computational time. In order to reduce these demanding computational costs, our approach combines a continuation technique and Newton's method with a Reduced Order Modeling (ROM) technique, suitably supplemented with a hyper-reduction method. To demonstrate the effectiveness of our ROM approach, we trace the steady solution branches of a nonlinear Schrödinger equation, called Gross-Pitaevskii equation, as one or two physical parameters are varied. In the two parameter study, we show that our approach is 60 times faster in constructing a bifurcation diagram than a standard Full Order Method.}
journal = {SIAM Journal on Scientific Computing},
}

20. F. Romor, M. Tezzele, A. Lario, and G. Rozza, "Kernel-based Active Subspaces with application to CFD parametric problems using Discontinuous Galerkin method", 2020.
A new method to perform a nonlinear reduction in parameter spaces is proposed. By using a kernel approach it is possible to find active subspaces in high-dimensional feature spaces. A mathematical foundation of the method is presented, with several applications to benchmark model functions, both scalar and vector-valued. We also apply the kernel-based active subspaces extension to a CFD parametric problem using the Discontinuous Galerkin method. A full comparison with respect to the linear active subspaces technique is provided for all the applications, proving the better performances of the proposed method. Moreover we show how the new kernel method overcomes the drawbacks of the active subspaces application for radial symmetric model functions.

@unpublished{RomorTezzeleLarioRozza2020,
author = {Francesco Romor and Marco Tezzele and Andrea Lario and Gianluigi Rozza},
title = {Kernel-based Active Subspaces with application to CFD parametric problems using Discontinuous Galerkin method},
year = {2020},
preprint = {https://arxiv.org/abs/2008.12083},
abstract = {A new method to perform a nonlinear reduction in parameter spaces is proposed. By using a kernel approach it is possible to find active subspaces in high-dimensional feature spaces. A mathematical foundation of the method is presented, with several applications to benchmark model functions, both scalar and vector-valued. We also apply the kernel-based active subspaces extension to a CFD parametric problem using the Discontinuous Galerkin method. A full comparison with respect to the linear active subspaces technique is provided for all the applications, proving the better performances of the proposed method. Moreover we show how the new kernel method overcomes the drawbacks of the active subspaces application for radial symmetric model functions.}
}

21. G. Rozza, M. Hess, G. Stabile, M. Tezzele, and F. Ballarin, "Basic Ideas and Tools for Projection-Based Model Reduction of Parametric Partial Differential Equations", in Handbook on Model Reduction, P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W. H. A. Schilders, and L. M. Silveira (eds.), , 2020.
We provide first the functional analysis background required for reduced order modeling and present the underlying concepts of reduced basis model reduction. The projection-based model reduction framework under affinity assumptions, offline-online decomposition and error estimation is introduced. Several tools for geometry parametrizations, such as free form deformation, radial basis function interpolation and inverse distance weighting interpolation are explained. The empirical interpolation method is introduced as a general tool to deal with non-affine parameter dependency and non-linear problems. The discrete and matrix versions of the empirical interpolation are considered as well. Active subspaces properties are discussed to reduce high-dimensional parameter spaces as a pre-processing step. Several examples illustrate the methodologies.

@inbook{RozzaHessStabileTezzeleBallarin2020,
author = {Gianluigi Rozza and Martin Hess and Giovanni Stabile and Marco Tezzele and Francesco Ballarin},
chapter = {Basic Ideas and Tools for Projection-Based Model Reduction of Parametric Partial Differential Equations},
year = {2020},
booktitle = {Handbook on Model Reduction},
editor = {P. Benner and S. Grivet-Talocia and A. Quarteroni and G. Rozza and W. H. A. Schilders and L. M. Silveira},
preprint = {https://arxiv.org/abs/1911.08954},
abstract = {We provide first the functional analysis background required for reduced order modeling and present the underlying concepts of reduced basis model reduction. The projection-based model reduction framework under affinity assumptions, offline-online decomposition and error estimation is introduced. Several tools for geometry parametrizations, such as free form deformation, radial basis function interpolation and inverse distance weighting interpolation are explained. The empirical interpolation method is introduced as a general tool to deal with non-affine parameter dependency and non-linear problems. The discrete and matrix versions of the empirical interpolation are considered as well. Active subspaces properties are discussed to reduce high-dimensional parameter spaces as a pre-processing step. Several examples illustrate the methodologies.}
}

22. G. Stabile, M. Zancanaro, and G. Rozza, "Efficient Geometrical parametrization for finite-volume based reduced order methods", International Journal for Numerical Methods in Engineering, 121(12), pp. 2655-2682, 2020.
In this work, we present an approach for the efficient treatment of parametrized geometries in the context of POD-Galerkin reduced order methods based on Finite Volume full order approximations. On the contrary to what is normally done in the framework of finite element reduced order methods, different geometries are not mapped to a common reference domain: the method relies on basis functions defined on an average deformed configuration and makes use of the Discrete Empirical Interpolation Method (D-EIM) to handle together non-affinity of the parametrization and non-linearities. In the first numerical example, different mesh motion strategies, based on a Laplacian smoothing technique and on a Radial Basis Function approach, are analyzed and compared on a heat transfer problem. Particular attention is devoted to the role of the non-orthogonal correction. In the second numerical example the methodology is tested on a geometrically parametrized incompressible Navier–Stokes problem. In this case, the reduced order model is constructed following the same segregated approach used at the full order level

@article{StabileZancanaroRozza2020,
author = {Giovanni Stabile and Matteo Zancanaro and Gianluigi Rozza},
journal = {International Journal for Numerical Methods in Engineering},
title = {Efficient Geometrical parametrization for finite-volume based reduced order methods},
year = {2020},
number = {12},
pages = {2655-2682},
volume = {121},
abstract = {In this work, we present an approach for the efficient treatment of parametrized geometries in the context of POD-Galerkin reduced order methods based on Finite Volume full order approximations. On the contrary to what is normally done in the framework of finite element reduced order methods, different geometries are not mapped to a common reference domain: the method relies on basis functions defined on an average deformed configuration and makes use of the Discrete Empirical Interpolation Method (D-EIM) to handle together non-affinity of the parametrization and non-linearities. In the first numerical example, different mesh motion strategies, based on a Laplacian smoothing technique and on a Radial Basis Function approach, are analyzed and compared on a heat transfer problem. Particular attention is devoted to the role of the non-orthogonal correction. In the second numerical example the methodology is tested on a geometrically parametrized incompressible Navier--Stokes problem. In this case, the reduced order model is constructed following the same segregated approach used at the full order level},
doi = {10.1002/nme.6324},
preprint = {https://arxiv.org/abs/1901.06373},
}

23. K. Star, G. Stabile, G. Rozza, and J. Degroote, "A POD-Galerkin reduced order model of a turbulent convective buoyant flow of sodium over a backward-facing step", 2020.
A Finite-Volume based POD-Galerkin reduced order modeling strategy for steady-state Reynolds averaged Navier–Stokes (RANS) simulation is extended for low-Prandtl number flow. The reduced order model is based on a full order model for which the effects of buoyancy on the flow and heat transfer are characterized by varying the Richardson number. The Reynolds stresses are computed with a linear eddy viscosity model. A single gradient diffusion hypothesis, together with a local correlation for the evaluation of the turbulent Prandtl number, is used to model the turbulent heat fluxes. The contribution of the eddy viscosity and turbulent thermal diffusivity fields are considered in the reduced order model with an interpolation based data-driven method. The reduced order model is tested for buoyancy-aided turbulent liquid sodium flow over a vertical backward-facing step with a uniform heat flux applied on the wall downstream of the step. The wall heat flux is incorporated with a Neumann boundary condition in both the full order model and the reduced order model. The velocity and temperature profiles predicted with the reduced order model for the same and new Richardson numbers inside the range of parameter values are in good agreement with the RANS simulations. Also, the local Stanton number and skin friction distribution at the heated wall are qualitatively well captured. Finally, the reduced order simulations, performed on a single core, are about $10^5$ times faster than the RANS simulations that are performed on eight cores.

@unpublished{StarStabileRozzaDegroote2020,
author = {Kelbij Star and Giovanni Stabile and Gianluigi Rozza and Joris Degroote},
title = {A POD-Galerkin reduced order model of a turbulent convective buoyant flow of sodium over a backward-facing step},
year = {2020},
abstract = {A Finite-Volume based POD-Galerkin reduced order modeling strategy for steady-state Reynolds averaged Navier--Stokes (RANS) simulation is extended for low-Prandtl number flow. The reduced order model is based on a full order model for which the effects of buoyancy on the flow and heat transfer are characterized by varying the Richardson number. The Reynolds stresses are computed with a linear eddy viscosity model. A single gradient diffusion hypothesis, together with a local correlation for the evaluation of the turbulent Prandtl number, is used to model the turbulent heat fluxes. The contribution of the eddy viscosity and turbulent thermal diffusivity fields are considered in the reduced order model with an interpolation based data-driven method. The reduced order model is tested for buoyancy-aided turbulent liquid sodium flow over a vertical backward-facing step with a uniform heat flux applied on the wall downstream of the step. The wall heat flux is incorporated with a Neumann boundary condition in both the full order model and the reduced order model. The velocity and temperature profiles predicted with the reduced order model for the same and new Richardson numbers inside the range of parameter values are in good agreement with the RANS simulations. Also, the local Stanton number and skin friction distribution at the heated wall are qualitatively well captured. Finally, the reduced order simulations, performed on a single core, are about $10^5$ times faster than the RANS simulations that are performed on eight cores.},
preprint = {https://arxiv.org/abs/2003.01114},
}

24. S. K. Star, B. Sanderse, G. Stabile, G. Rozza, and J. Degroote, "Reduced order models for the incompressible Navier-Stokes equations on collocated grids using a 'discretize-then-project' approach", 2020.
A novel reduced order model (ROM) for incompressible flows is developed by performing a Galerkin projection based on a fully (space and time) discrete full order model (FOM) formulation. This 'discretize-then-project' approach requires no pressure stabilization technique (even though the pressure term is present in the ROM) nor a boundary control technique (to impose the boundary conditions at the ROM level). These are two main advantages compared to existing approaches. The fully discrete FOM is obtained by a finite volume discretization of the incompressible Navier-Stokes equations on a collocated grid, with a forward Euler time discretization. Two variants of the time discretization method, the inconsistent and consistent flux method, have been investigated. The latter leads to divergence-free velocity fields, also on the ROM level, whereas the velocity fields are only approximately divergence-free in the former method. For both methods, stable and accurate results have been obtained for test cases with different types of boundary conditions: a lid-driven cavity and an open-cavity (with an inlet and outlet). The ROM obtained with the consistent flux method, having divergence-free velocity fields, is slightly more accurate but also slightly more expensive to solve compared to the inconsistent flux method. The speedup ratio of the ROM and FOM computation times is the highest for the open cavity test case with the inconsistent flux method.

@unpublished{StarSanderseStabileRozzaDegroote2020,
author = {Sabrina Kelbij Star and Benjamin Sanderse and Giovanni Stabile and Gianluigi Rozza and Joris Degroote},
title = {Reduced order models for the incompressible Navier-Stokes equations on collocated grids using a 'discretize-then-project' approach},
year = {2020},
preprint = {https://arxiv.org/abs/2010.04953},
abstract = {A novel reduced order model (ROM) for incompressible flows is developed by performing a Galerkin projection based on a fully (space and time) discrete full order model (FOM) formulation. This 'discretize-then-project' approach requires no pressure stabilization technique (even though the pressure term is present in the ROM) nor a boundary control technique (to impose the boundary conditions at the ROM level). These are two main advantages compared to existing approaches. The fully discrete FOM is obtained by a finite volume discretization of the incompressible Navier-Stokes equations on a collocated grid, with a forward Euler time discretization. Two variants of the time discretization method, the inconsistent and consistent flux method, have been investigated. The latter leads to divergence-free velocity fields, also on the ROM level, whereas the velocity fields are only approximately divergence-free in the former method. For both methods, stable and accurate results have been obtained for test cases with different types of boundary conditions: a lid-driven cavity and an open-cavity (with an inlet and outlet). The ROM obtained with the consistent flux method, having divergence-free velocity fields, is slightly more accurate but also slightly more expensive to solve compared to the inconsistent flux method. The speedup ratio of the ROM and FOM computation times is the highest for the open cavity test case with the inconsistent flux method.}
}

25. M. Strazzullo, Z. Zainib, F. Ballarin, and G. Rozza, "Reduced order methods for parametrized non-linear and time dependent optimal flow control problems, towards applications in biomedical and environmental sciences", in ENUMATH2019 proceedings, 2020.
We introduce reduced order methods as an efficient strategy to solve parametrized non-linear and time dependent optimal flow control problems governed by partial differential equations. Indeed, optimal control problems require a huge computational effort in order to be solved, most of all in a physical and/or geometrical parametrized setting. Reduced order methods are a reliably suitable approach, increasingly gaining popularity, to achieve rapid and accurate optimal solutions in several fields, such as in biomedical and environmental sciences. In this work, we exploit POD-Galerkin reduction over a parametrized optimality system, derived from Karush-Kuhn-Tucker conditions. The methodology presented is tested on two boundary control problems, governed respectively by (i) time dependent Stokes equations and (ii) steady non-linear Navier-Stokes equations.

@inproceedings{StrazzulloZainibBallarinRozza2020,
author = {Maria Strazzullo and Zakia Zainib and Francesco Ballarin and Gianluigi Rozza},
title = {Reduced order methods for parametrized non-linear and time dependent optimal flow control problems, towards applications in biomedical and environmental sciences},
year = {2020},
booktitle = {ENUMATH2019 proceedings},
publisher = {Springer},
preprint = {https://arxiv.org/abs/1912.07886},
abstract = {We introduce reduced order methods as an efficient strategy to solve parametrized non-linear and time dependent optimal flow control problems governed by partial differential equations. Indeed, optimal control problems require a huge computational effort in order to be solved, most of all in a physical and/or geometrical parametrized setting. Reduced order methods are a reliably suitable approach, increasingly gaining popularity, to achieve rapid and accurate optimal solutions in several fields, such as in biomedical and environmental sciences. In this work, we exploit POD-Galerkin reduction over a parametrized optimality system, derived from Karush-Kuhn-Tucker conditions. The methodology presented is tested on two boundary control problems, governed respectively by (i) time dependent Stokes equations and (ii) steady non-linear Navier-Stokes equations.}
}

26. M. Strazzullo, F. Ballarin, and G. Rozza, "POD-Galerkin Model Order Reduction for Parametrized Nonlinear Time Dependent Optimal Flow Control: an Application to Shallow Water Equations", 2020.
In this work we propose reduced order methods as a reliable strategy to efficiently solve parametrized optimal control problems governed by shallow waters equations in a solution tracking setting. The physical parametrized model we deal with is nonlinear and time dependent: this leads to very time consuming simulations which can be unbearable e.g. in a marine environmental monitoring plan application. Our aim is to show how reduced order modelling could help in studying different configurations and phenomena in a fast way. After building the optimality system, we rely on a POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space. The presented theoretical framework is actually suited to general nonlinear time dependent optimal control problems. The proposed methodology is finally tested with a numerical experiment: the reduced optimal control problem governed by shallow waters equations reproduces the desired velocity and height profiles faster than the standard model, still remaining accurate.

@unpublished{StrazzulloBallarinRozza2020,
author = {Maria Strazzullo and Francesco Ballarin and Gianluigi Rozza},
title = {POD-Galerkin Model Order Reduction for Parametrized Nonlinear Time Dependent Optimal Flow Control: an Application to Shallow Water Equations},
year = {2020},
preprint = {https://arxiv.org/abs/2003.09695},
abstract = {In this work we propose reduced order methods as a reliable strategy to efficiently solve parametrized optimal control problems governed by shallow waters equations in a solution tracking setting. The physical parametrized model we deal with is nonlinear and time dependent: this leads to very time consuming simulations which can be unbearable e.g. in a marine environmental monitoring plan application. Our aim is to show how reduced order modelling could help in studying different configurations and phenomena in a fast way. After building the optimality system, we rely on a POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space. The presented theoretical framework is actually suited to general nonlinear time dependent optimal control problems. The proposed methodology is finally tested with a numerical experiment: the reduced optimal control problem governed by shallow waters equations reproduces the desired velocity and height profiles faster than the standard model, still remaining accurate.}
}

27. M. Strazzullo, F. Ballarin, and G. Rozza, "POD–Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation", Journal of Scientific Computing, 83(3), pp. 55, 2020.
@article{StrazzulloBallarinRozza2020,
author = {Maria Strazzullo and Francesco Ballarin and Gianluigi Rozza},
title = {POD--Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation},
journal = {Journal of Scientific Computing},
volume = {83},
number = {3},
pages = {55},
year = {2020},
preprint = {https://arxiv.org/abs/1909.09631},
doi = {10.1007/s10915-020-01232-x}
abstract = {In this work we deal with parametrized time dependent optimal control problems governed by partial differential equations. We aim at extending the standard saddle point framework of steady constraints to time dependent cases. We provide an analysis of the well-posedness of this formulation both for parametrized scalar parabolic constraint and Stokes governing equations and we propose reduced order methods as an effective strategy to solve them. Indeed, on one hand, parametrized time dependent optimal control is a very powerful mathematical model which is able to describe several physical phenomena, on the other, it requires a huge computational effort. Reduced order methods are a suitable approach to have rapid and accurate simulations. We rely on POD–Galerkin reduction over the physical and geometrical parameters of the optimality system in a space-time formulation. Our theoretical results and our methodology are tested on two examples: a boundary time dependent optimal control for a Graetz flow and a distributed optimal control governed by time dependent Stokes equations. With these two test cases the convenience of the reduced order modelling is further extended to the field of time dependent optimal control.}
}

28. M. Tezzele, N. Demo, G. Stabile, A. Mola, and G. Rozza, "Enhancing CFD predictions in shape design problems by model and parameter space reduction", 2020.
In this work we present an advanced computational pipeline for the approximation and prediction of the lift coefficient of a parametrized airfoil profile. The non-intrusive reduced order method is based on dynamic mode decomposition (DMD) and it is coupled with dynamic active subspaces (DyAS) to enhance the future state prediction of the target function and reduce the parameter space dimensionality. The pipeline is based on high-fidelity simulations carried out by the application of finite volume method for turbulent flows, and automatic mesh morphing through radial basis functions interpolation technique. The proposed pipeline is able to save 1/3 of the overall computational resources thanks to the application of DMD. Moreover exploiting DyAS and performing the regression on a lower dimensional space results in the reduction of the relative error in the approximation of the time-varying lift coefficient by a factor 2 with respect to using only the DMD.

@unpublished{TezzeleDemoStabileMolaRozza2020,
author = {Marco Tezzele and Nicola Demo and Giovanni Stabile and Andrea Mola and Gianluigi Rozza},
title = {Enhancing CFD predictions in shape design problems by model and parameter space reduction},
year = {2020},
abstract = {In this work we present an advanced computational pipeline for the approximation and prediction of the lift coefficient of a parametrized airfoil profile. The non-intrusive reduced order method is based on dynamic mode decomposition (DMD) and it is coupled with dynamic active subspaces (DyAS) to enhance the future state prediction of the target function and reduce the parameter space dimensionality. The pipeline is based on high-fidelity simulations carried out by the application of finite volume method for turbulent flows, and automatic mesh morphing through radial basis functions interpolation technique. The proposed pipeline is able to save 1/3 of the overall computational resources thanks to the application of DMD. Moreover exploiting DyAS and performing the regression on a lower dimensional space results in the reduction of the relative error in the approximation of the time-varying lift coefficient by a factor 2 with respect to using only the DMD.},
preprint = {https://arxiv.org/abs/2001.05237},
}

29. Z. Zainib, F. Ballarin, S. Fremes, P. Triverio, L. Jiménez-Juan, and G. Rozza, "Reduced order methods for parametric optimal flow control in coronary bypass grafts, towards patient-specific data assimilation", International Journal for Numerical Methods in Biomedical Engineering, 2020.
Coronary artery bypass grafts (CABG) surgery is an invasive procedure performed to circumvent partial or complete blood flow blockage in coronary artery disease (CAD). In this work, we apply a numerical optimal flow control model to patient-specific geometries of CABG, reconstructed from clinical images of real-life surgical cases, in parameterized settings. The aim of these applications is to match known physiological data with numerical hemodynamics corresponding to different scenarios, arisen by tuning some parameters. Such applications are an initial step towards matching patient-specific physiological data in patient-specific vascular geometries as best as possible. Two critical challenges that reportedly arise in such problems are, (i). lack of robust quantification of meaningful boundary conditions required to match known data as best as possible and (ii). high computational cost. In this work, we utilize unknown control variables in the optimal flow control problems to take care of the first challenge. Moreover, to address the second challenge, we propose a time-efficient and reliable computational environment for such parameterized problems by projecting them onto a low-dimensional solution manifold through proper orthogonal decomposition (POD)–Galerkin.

@article{ZainibBallarinFremesTriverioJimenezJuanRozza2020,
author = {Zakia Zainib and Francesco Ballarin and Stephen Fremes and Piero Triverio and Laura Jiménez-Juan and Gianluigi Rozza},
title = {Reduced order methods for parametric optimal flow control in coronary bypass grafts, towards patient-specific data assimilation},
year = {2020},
preprint = {https://arxiv.org/abs/1911.01409},
doi = {10.1002/cnm.3367},
journal = {International Journal for Numerical Methods in Biomedical Engineering},
abstract = {Coronary artery bypass grafts (CABG) surgery is an invasive procedure performed to circumvent partial or complete blood flow blockage in coronary artery disease (CAD). In this work, we apply a numerical optimal flow control model to patient-specific geometries of CABG, reconstructed from clinical images of real-life surgical cases, in parameterized settings. The aim of these applications is to match known physiological data with numerical hemodynamics corresponding to different scenarios, arisen by tuning some parameters. Such applications are an initial step towards matching patient-specific physiological data in patient-specific vascular geometries as best as possible.
Two critical challenges that reportedly arise in such problems are, (i). lack of robust quantification of meaningful boundary conditions required to match known data as best as possible and (ii). high computational cost. In this work, we utilize unknown control variables in the optimal flow control problems to take care of the first challenge. Moreover, to address the second challenge, we propose a time-efficient and reliable computational environment for such parameterized problems by projecting them onto a low-dimensional solution manifold through proper orthogonal decomposition (POD)--Galerkin.}
}

### 2019

1. S. Busto, G. Stabile, G. Rozza, and M. E. Vázquez-Cendón, "POD-Galerkin reduced order methods for combined Navier-Stokes transport equations based on a hybrid FV-FE solver", Computers & Mathematics with Applications, 2019.
The purpose of this work is to introduce a novel POD-Galerkin strategy for the hybrid finite volume/finite element solver introduced in Bermúdez et al. 2014 and Busto et al. 2018. The interest is into the incompressible Navier-Stokes equations coupled with an additional transport equation. The full order model employed in this article makes use of staggered meshes. This feature will be conveyed to the reduced order model leading to the definition of reduced basis spaces in both meshes. The reduced order model presented herein accounts for velocity, pressure, and a transport-related variable. The pressure term at both the full order and the reduced order level is reconstructed making use of a projection method. More precisely, a Poisson equation for pressure is considered within the reduced order model. Results are verified against three-dimensional manufactured test cases. Moreover a modified version of the classical cavity test benchmark including the transport of a species is analysed.

@article{BustoStabileRozzaCendon2019,
author = {S. Busto and G. Stabile and G. Rozza and M.E. V{\'{a}}zquez-Cend{\'{o}}n},
journal = {Computers {\&} Mathematics with Applications},
title = {POD-Galerkin reduced order methods for combined Navier-Stokes transport equations based on a hybrid {FV}-{FE} solver},
year = {2019},
abstract = {The purpose of this work is to introduce a novel POD-Galerkin strategy for the hybrid finite volume/finite element solver introduced in Bermúdez et al. 2014 and Busto et al. 2018. The interest is into the incompressible Navier-Stokes equations coupled with an additional transport equation. The full order model employed in this article makes use of staggered meshes. This feature will be conveyed to the reduced order model leading to the definition of reduced basis spaces in both meshes. The reduced order model presented herein accounts for velocity, pressure, and a transport-related variable. The pressure term at both the full order and the reduced order level is reconstructed making use of a projection method. More precisely, a Poisson equation for pressure is considered within the reduced order model. Results are verified against three-dimensional manufactured test cases. Moreover a modified version of the classical cavity test benchmark including the transport of a species is analysed.},
doi = {10.1016/j.camwa.2019.06.026},
preprint = {https://arxiv.org/abs/1810.07999}
}

2. N. Demo, M. Tezzele, and G. Rozza, "A non-intrusive approach for the reconstruction of POD modal coefficients through active subspaces", Comptes Rendus Mécanique, 2019.
Reduced order modeling (ROM) provides an efficient framework to compute solutions of parametric problems. Basically, it exploits a set of precomputed high-fidelity solutions –- computed for properly chosen parameters, using a full-order model –- in order to find the low dimensional space that contains the solution manifold. Using this space, an approximation of the numerical solution for new parameters can be computed in real-time response scenario, thanks to the reduced dimensionality of the problem. In a ROM framework, the most expensive part from the computational viewpoint is the calculation of the numerical solutions using the full-order model. Of course, the number of collected solutions is strictly related to the accuracy of the reduced order model. In this work, we aim at increasing the precision of the model also for few input solutions by coupling the proper orthogonal decomposition with interpolation (PODI) –- a data-driven reduced order method –- with the active subspace (AS) property, an emerging tool for reduction in parameter space. The enhanced ROM results in a reduced number of input solutions to reach the desired accuracy. In this contribution, we present the numerical results obtained by applying this method to a structural problem and in a fluid dynamics one.

@article{DemoTezzeleRozza2019,
author = {Demo, Nicola and Tezzele, Marco and Rozza, Gianluigi},
title = {A non-intrusive approach for the reconstruction of POD modal coefficients through active subspaces},
journal = {Comptes Rendus Mécanique},
doi = {10.1016/j.crme.2019.11.012},
year = {2019},
preprint = {https://arxiv.org/abs/1907.12777},
abstract = {Reduced order modeling (ROM) provides an efficient framework to compute solutions of parametric problems. Basically, it exploits a set of precomputed high-fidelity solutions --- computed for properly chosen parameters, using a full-order model --- in order to find the low dimensional space that contains the solution manifold. Using this space, an approximation of the numerical solution for new parameters can be computed in real-time response scenario, thanks to the reduced dimensionality of the problem. In a ROM framework, the most expensive part from the computational viewpoint is the calculation of the numerical solutions using the full-order model. Of course, the number of collected solutions is strictly related to the accuracy of the reduced order model. In this work, we aim at increasing the precision of the model also for few input solutions by coupling the proper orthogonal decomposition with interpolation (PODI) --- a data-driven reduced order method --- with the active subspace (AS) property, an emerging tool for reduction in parameter space. The enhanced ROM results in a reduced number of input solutions to reach the desired accuracy. In this contribution, we present the numerical results obtained by applying this method to a structural problem and in a fluid dynamics one.}
}

3. N. Demo, M. Tezzele, A. Mola, and G. Rozza, "A complete data-driven framework for the efficient solution of parametric shape design and optimisation in naval engineering problems", in VIII International Conference on Computational Methods in Marine Engineering, 2019.
In the reduced order modeling (ROM) framework, the solution of a parametric partial differential equation is approximated by combining the high-fidelity solutions of the problem at hand for several properly chosen configurations. Examples of the ROM application, in the naval field, can be found in [31, 24]. Mandatory ingredient for the ROM methods is the relation between the high-fidelity solutions and the parameters. Dealing with geometrical parameters, especially in the industrial context, this relation may be unknown and not trivial (simulations over hand morphed geometries) or very complex (high number of parameters or many nested morphing techniques). To overcome these scenarios, we propose in this contribution an efficient and complete data-driven framework involving ROM techniques for shape design and optimization, extending the pipeline presented in [7]. By applying the singular value decomposition (SVD) to the points coordinates defining the hull geometry –- assuming the topology is inaltered by the deformation –-, we are able to compute the optimal space which the deformed geometries belong to, hence using the modal coefficients as the new parameters we can reconstruct the parametric formulation of the domain. Finally the output of interest is approximated using the proper orthogonal decomposition with interpolation technique. To conclude, we apply this framework to a naval shape design problem where the bulbous bow is morphed to reduce the total resistance of the ship advancing in calm water.

@inproceedings{DemoTezzeleMolaRozza2019,
author = {Demo, Nicola and Tezzele, Marco and Mola, Andrea and Rozza, Gianluigi},
title = {A complete data-driven framework for the efficient solution of parametric shape design and optimisation in naval engineering problems},
year = {2019},
booktitle = {VIII International Conference on Computational Methods in Marine Engineering},
preprint = {https://arxiv.org/abs/1905.05982},
abstract = {In the reduced order modeling (ROM) framework, the solution of a parametric partial differential equation is approximated by combining the high-fidelity solutions of the problem at hand for several properly chosen configurations. Examples of the ROM application, in the naval field, can be found in [31, 24]. Mandatory ingredient for the ROM methods is the relation between the high-fidelity solutions and the parameters. Dealing with geometrical parameters, especially in the industrial context, this relation may be unknown and not trivial (simulations over hand morphed geometries) or very complex (high number of parameters or many nested morphing techniques). To overcome these scenarios, we propose in this contribution an efficient and complete data-driven framework involving ROM techniques for shape design and optimization, extending the pipeline presented in [7]. By applying the singular value decomposition (SVD) to the points coordinates defining the hull geometry --- assuming the topology is inaltered by the deformation ---, we are able to compute the optimal space which the deformed geometries belong to, hence using the modal coefficients as the new parameters we can reconstruct the parametric formulation of the domain. Finally the output of interest is approximated using the proper orthogonal decomposition with interpolation technique. To conclude, we apply this framework to a naval shape design problem where the bulbous bow is morphed to reduce the total resistance of the ship advancing in calm water.}
}

4. M. Gadalla, M. Tezzele, A. Mola, and G. Rozza, "BladeX: Python Blade Morphing", The Journal of Open Source Software, 4(34), pp. 1203, 2019.
@article{GadallaTezzeleMolaRozza2019,
author = {Gadalla, Mahmoud and Tezzele, Marco and Mola, Andrea and Rozza, Gianluigi},
journal = {The Journal of Open Source Software},
preprint = {https://www.theoj.org/joss-papers/joss.01203/10.21105.joss.01203.pdf},
volume = {4},
number = {34},
pages = {1203},
year = {2019},
doi = {10.21105/joss.01203}
}

5. S. Georgaka, G. Stabile, G. Rozza, and M. J. Bluck, "Parametric POD-Galerkin Model Order Reduction for Unsteady-State Heat Transfer Problems", Communications in Computational Physics, 27(1), pp. 1–32, 2019.
A parametric reduced order model based on proper orthogonal decom- position with Galerkin projection has been developed and applied for the modeling of heat transport in T-junction pipes which are widely found in nuclear power plants. Thermal mixing of different temperature coolants in T-junction pipes leads to tem- perature fluctuations and this could potentially cause thermal fatigue in the pipe walls. The novelty of this paper is the development of a parametric ROM considering the three dimensional, incompressible, unsteady Navier-Stokes equations coupled with the heat transport equation in a finite volume approximation. Two different paramet- ric cases are presented in this paper: parametrization of the inlet temperatures and parametrization of the kinematic viscosity. Different training spaces are considered and the results are compared against the full order model.

@article{GeorgakaStabileRozzaBluck2019,
author = {Sokratia Georgaka and Giovanni Stabile and Gianluigi Rozza and Michael J. Bluck},
journal = {Communications in Computational Physics},
title = {Parametric POD-Galerkin Model Order Reduction for Unsteady-State Heat Transfer Problems},
year = {2019},
issn = {1991-7120},
number = {1},
pages = {1--32},
volume = {27},
abstract = {A parametric reduced order model based on proper orthogonal decom- position with Galerkin projection has been developed and applied for the modeling of heat transport in T-junction pipes which are widely found in nuclear power plants. Thermal mixing of different temperature coolants in T-junction pipes leads to tem- perature fluctuations and this could potentially cause thermal fatigue in the pipe walls. The novelty of this paper is the development of a parametric ROM considering the three dimensional, incompressible, unsteady Navier-Stokes equations coupled with the heat transport equation in a finite volume approximation. Two different paramet- ric cases are presented in this paper: parametrization of the inlet temperatures and parametrization of the kinematic viscosity. Different training spaces are considered and the results are compared against the full order model.},
doi = {10.4208/cicp.OA-2018-0207},
preprint = {https://arxiv.org/abs/1808.05175}
}

6. M. Girfoglio, A. Quaini, and G. Rozza, "A Finite Volume approximation of the Navier-Stokes equations with nonlinear filtering stabilization", Computers & Fluids, 187, pp. 27-45, 2019.
We consider a Leray model with a nonlinear differential low-pass filter for the simulation of incompressible fluid flow at moderately large Reynolds number (in the range of a few thousands) with under-refined meshes. For the implementation of the model, we adopt the three-step algorithm Evolve-Filter-Relax (EFR). The Leray model has been extensively applied within a Finite Element (FE) framework. Here, we propose to combine the EFR algorithm with a computationally efficient Finite Volume (FV) method. Our approach is validated against numerical data available in the literature for the 2D flow past a cylinder and against experimental measurements for the 3D fluid flow in an idealized medical device, as recommended by the U.S. Food and Drug Administration. We will show that for similar levels of mesh refinement FV and FE methods provide significantly different results. Through our numerical experiments, we are able to provide practical directions to tune the parameters involved in the model. Furthermore, we are able to investigate the impact of mesh features (element type, non-orthogonality, local refinement, and element aspect ratio) and the discretization method for the convective term on the agreement between numerical solutions and experimental data.

@article{GirfoglioQuainiRozza2019,
author = {Girfoglio, Michele and Quaini, Annalisa and Rozza, Gianluigi},
title = {A Finite Volume approximation of the Navier-Stokes equations with nonlinear filtering stabilization},
preprint = {https://arxiv.org/abs/1901.05251},
journal = {Computers \& Fluids},
volume = {187},
pages = {27-45},
year = {2019},
doi = {10.1016/j.compfluid.2019.05.001},
abstract = {We consider a Leray model with a nonlinear differential low-pass filter for the simulation of incompressible fluid flow at moderately large Reynolds number (in the range of a few thousands) with under-refined meshes. For the implementation of the model, we adopt the three-step algorithm Evolve-Filter-Relax (EFR). The Leray model has been extensively applied within a Finite Element (FE) framework. Here, we propose to combine the EFR algorithm with a computationally efficient Finite Volume (FV) method. Our approach is validated against numerical data available in the literature for the 2D flow past a cylinder and against experimental measurements for the 3D fluid flow in an idealized medical device, as recommended by the U.S. Food and Drug Administration. We will show that for similar levels of mesh refinement FV and FE methods provide significantly different results. Through our numerical experiments, we are able to provide practical directions to tune the parameters involved in the model. Furthermore, we are able to investigate the impact of mesh features (element type, non-orthogonality, local refinement, and element aspect ratio) and the discretization method for the convective term on the agreement between numerical solutions and experimental data.}
}

7. M. W. Hess and G. Rozza, "A Spectral Element Reduced Basis Method in Parametric CFD", in Numerical Mathematics and Advanced Applications - ENUMATH 2017, F. A. Radu, K. Kumar, I. Berre, J. M. Nordbotten, and I. S. Pop (eds.), Springer International Publishing, vol. 126, 2019.
We consider the Navier-Stokes equations in a channel with varying Reynolds numbers. The model is discretized with high-order spectral element ansatz functions, resulting in 14 259 degrees of freedom. The steady-state snapshot solu- tions define a reduced order space, which allows to accurately evaluate the steady- state solutions for varying Reynolds number with a reduced order model within a fixed-point iteration. In particular, we compare different aspects of implementing the reduced order model with respect to the use of a spectral element discretization. It is shown, how a multilevel static condensation in the pressure and velocity boundary degrees of freedom can be combined with a reduced order modelling approach to enhance computational times in parametric many-query scenarios.

@inbook{HessRozza2019,
author = {Hess, Martin W. and Rozza, Gianluigi},
editor = {Radu, Florin Adrian and Kumar, Kundan and Berre, Inga and Nordbotten, Jan Martin and Pop, Iuliu Sorin},
year = {2019},
chapter = {A Spectral Element Reduced Basis Method in Parametric CFD},
booktitle = {Numerical Mathematics and Advanced Applications - ENUMATH 2017},
volume = {126},
doi = {10.1007/978-3-319-96415-7_64}
pages = {693--701},
publisher = {Springer International Publishing},
preprint = {https://arxiv.org/abs/1712.06432},
abstract = {We consider the Navier-Stokes equations in a channel with varying Reynolds numbers. The model is discretized with high-order spectral element ansatz functions, resulting in 14 259 degrees of freedom. The steady-state snapshot solu- tions define a reduced order space, which allows to accurately evaluate the steady- state solutions for varying Reynolds number with a reduced order model within a fixed-point iteration. In particular, we compare different aspects of implementing the reduced order model with respect to the use of a spectral element discretization. It is shown, how a multilevel static condensation in the pressure and velocity boundary degrees of freedom can be combined with a reduced order modelling approach to enhance computational times in parametric many-query scenarios.}
}

8. M. Hess, A. Alla, A. Quaini, G. Rozza, and M. Gunzburger, "A Localized Reduced-Order Modeling Approach for PDEs with Bifurcating Solutions", Computer Methods in Applied Mechanics and Engineering, 351, pp. 379-403, 2019.
Reduced-order modeling (ROM) commonly refers to the construction, based on a few solutions (referred to as snapshots) of an expensive discretized partial differential equation (PDE), and the subsequent application of low-dimensional discretizations of partial differential equations (PDEs) that can be used to more efficiently treat problems in control and optimization, uncertainty quantification, and other settings that require multiple approximate PDE solutions. In this work, a ROM is developed and tested for the treatment of nonlinear PDEs whose solutions bifurcate as input parameter values change. In such cases, the parameter domain can be subdivided into subregions, each of which corresponds to a different branch of solutions. Popular ROM approaches such as proper orthogonal decomposition (POD), results in a global low-dimensional basis that does no respect not take advantage of the often large differences in the PDE solutions corresponding to different subregions. Instead, in the new method, the k-means algorithm is used to cluster snapshots so that within cluster snapshots are similar to each other and are dissimilar to those in other clusters. This is followed by the construction of local POD bases, one for each cluster. The method also can detect which cluster a new parameter point belongs to, after which the local basis corresponding to that cluster is used to determine a ROM approximation. Numerical experiments show the effectiveness of the method both for problems for which bifurcation cause continuous and discontinuous changes in the solution of the PDE.

@article{HessAllaQuainiRozzaGunzburger2019,
author = {Hess, Martin and Alla, Alessandro and Quaini, Annalisa and Rozza, Gianluigi and Gunzburger, Max},
title = {A Localized Reduced-Order Modeling Approach for PDEs with Bifurcating Solutions},
journal = {Computer Methods in Applied Mechanics and Engineering},
volume = {351},
pages = {379-403},
year = {2019},
doi = {10.1016/j.cma.2019.03.050},
preprint = {https://arxiv.org/abs/1807.08851},
abstract = {Reduced-order modeling (ROM) commonly refers to the construction, based on a few solutions (referred to as snapshots) of an expensive discretized partial differential equation (PDE), and the subsequent application of low-dimensional discretizations of partial differential equations (PDEs) that can be used to more efficiently treat problems in control and optimization, uncertainty quantification, and other settings that require multiple approximate PDE solutions. In this work, a ROM is developed and tested for the treatment of nonlinear PDEs whose solutions bifurcate as input parameter values change. In such cases, the parameter domain can be subdivided into subregions, each of which corresponds to a different branch of solutions. Popular ROM approaches such as proper orthogonal decomposition (POD), results in a global low-dimensional basis that does no respect not take advantage of the often large differences in the PDE solutions corresponding to different subregions. Instead, in the new method, the k-means algorithm is used to cluster snapshots so that within cluster snapshots are similar to each other and are dissimilar to those in other clusters. This is followed by the construction of local POD bases, one for each cluster. The method also can detect which cluster a new parameter point belongs to, after which the local basis corresponding to that cluster is used to determine a ROM approximation. Numerical experiments show the effectiveness of the method both for problems for which bifurcation cause continuous and discontinuous changes in the solution of the PDE.}
}

9. E. N. Karatzas, G. Stabile, L. Nouveau, G. Scovazzi, and G. Rozza, "A Reduced-Order Shifted Boundary Method for Parametrized incompressible Navier-Stokes equations", 2019.
We investigate a projection-based reduced-order model of the steady incompressible Navier-Stokes equations for moderate Reynolds numbers. In particular, we construct an "embedded" reduced basis space, by applying proper orthogonal decomposition to the Shifted Boundary Method, a high-fidelity embedded method recently developed. We focus on the geometrical parametrization through level-set geometries, using a fixed Cartesian background geometry and the associated mesh. This approach avoids both remeshing and the development of a reference domain formulation, as typically done in fitted mesh finite element formulations. Two-dimensional computational examples for one and three-parameter dimensions are presented to validate the convergence and the efficacy of the proposed approach.

@unpublished{KaratzasStabileNouveauScovazziRozza2019,
author = {Karatzas, Efthymios N. and Stabile, Giovanni and Nouveau, Leo and Scovazzi, Guglielmo and Rozza, Gianluigi},
title = {A Reduced-Order Shifted Boundary Method for Parametrized incompressible Navier-Stokes equations},
year = {2019},
preprint = {https://arxiv.org/abs/1907.10549},
abstract = {We investigate a projection-based reduced-order model of the steady incompressible Navier-Stokes equations for moderate Reynolds numbers. In particular, we construct an "embedded" reduced basis space, by applying proper orthogonal decomposition to the Shifted Boundary Method, a high-fidelity embedded method recently developed. We focus on the geometrical parametrization through level-set geometries, using a fixed Cartesian background geometry and the associated mesh. This approach avoids both remeshing and the development of a reference domain formulation, as typically done in fitted mesh finite element formulations. Two-dimensional computational examples for one and three-parameter dimensions are presented to validate the convergence and the efficacy of the proposed approach.}
}

10. E. N. Karatzas, G. Stabile, L. Nouveau, G. Scovazzi, and G. Rozza, "A reduced basis approach for PDEs on parametrized geometries based on the shifted boundary finite element method and application to a Stokes flow", Computer Methods in Applied Mechanics and Engineering, 347, pp. 568–587, 2019.
We propose a model order reduction technique integrating the Shifted Boundary Method (SBM) with a POD-Galerkin strategy. This approach allows to treat more complex parametrized domains in an efficient and straightforward way. The impact of the proposed approach is threefold. First, problems involving parametrizations of complex geometrical shapes and/or large domain deformations can be efficiently solved at full-order by means of the SBM, an unfitted boundary method that avoids remeshing and the tedious handling of cut cells by introducing an approximate surrogate boundary. Second, the computational effort is further reduced by the development of a reduced order model (ROM) technique based on a POD-Galerkin approach. Third, the SBM provides a smooth mapping from the true to the surrogate domain, and for this reason, the stability and performance of the reduced order basis are enhanced. This feature is the net result of the combination of the proposed ROM approach and the SBM. Similarly, the combination of the SBM with a projection-based ROM gives the great advantage of an easy and fast to implement algorithm considering geometrical parametrization with large deformations. The transformation of each geometry to a reference geometry (morphing) is in fact not required. These combined advantages will allow the solution of PDE problems more efficiently. We illustrate the performance of this approach on a number of two-dimensional Stokes flow problems.

@article{KaratzasStabileNouveauScovazziRozza2019,
author = {Karatzas, Efthymios N and Stabile, Giovanni and Nouveau, Leo and Scovazzi, Guglielmo and Rozza, Gianluigi},
journal = {Computer Methods in Applied Mechanics and Engineering},
title = {A reduced basis approach for PDEs on parametrized geometries based on the shifted boundary finite element method and application to a Stokes flow},
year = {2019},
pages = {568--587},
volume = {347},
abstract = {We propose a model order reduction technique integrating the Shifted Boundary Method (SBM) with a POD-Galerkin strategy. This approach allows to treat more complex parametrized domains in an efficient and straightforward way. The impact of the proposed approach is threefold.
First, problems involving parametrizations of complex geometrical shapes and/or large domain deformations can be efficiently solved at full-order by means of the SBM, an unfitted boundary method that avoids remeshing and the tedious handling of cut cells by introducing an approximate surrogate boundary.
Second, the computational effort is further reduced by the development of a reduced order model (ROM) technique based on a POD-Galerkin approach.
Third, the SBM provides a smooth mapping from the true to the surrogate domain, and for this reason, the stability and performance of the reduced order basis are enhanced. This feature is the net result of the combination of the proposed ROM approach and the SBM. Similarly, the combination of the SBM with a projection-based ROM gives the great advantage of an easy and fast to implement algorithm considering geometrical parametrization with large deformations. The transformation of each geometry to a reference geometry (morphing) is in fact not required.
These combined advantages will allow the solution of PDE problems more efficiently. We illustrate the performance of this approach on a number of two-dimensional Stokes flow problems.},
doi = {10.1016/j.cma.2018.12.040},
preprint = {https://arxiv.org/abs/1807.07790},
}

11. A. Mola, M. Tezzele, M. Gadalla, F. Valdenazzi, D. Grassi, R. Padovan, and G. Rozza, "Efficient Reduction in Shape Parameter Space Dimension for Ship Propeller Blade Design", in VIII International Conference on Computational Methods in Marine Engineering, 2019.
In this work, we present the results of a ship propeller design optimization campaign carried out in the framework of the research project PRELICA, funded by the Friuli Venezia Giulia regional government. The main idea of this work is to operate on a multidisciplinary level to identify propeller shapes that lead to reduced tip vortex-induced pressure and increased efficiency without altering the thrust. First, a specific tool for the bottom-up construction of parameterized propeller blade geometries has been developed. The algorithm proposed operates with a user defined number of arbitrary shaped or NACA airfoil sections, and employs arbitrary degree NURBS to represent the chord, pitch, skew and rake distribution as a function of the blade radial coordinate. The control points of such curves have been modified to generate, in a fully automated way, a family of blade geometries depending on as many as 20 shape parameters. Such geometries have then been used to carry out potential flow simulations with the Boundary Element Method based software PROCAL. Given the high number of parameters considered, such a preliminary stage allowed for a fast evaluation of the performance of several hundreds of shapes. In addition, the data obtained from the potential flow simulation allowed for the application of a parameter space reduction methodology based on active subspaces (AS) property, which suggested that the main propeller performance indices are, at a first but rather accurate approximation, only depending on a single parameter which is a linear combination of all the original geometric ones. AS analysis has also been used to carry out a constrained optimization exploiting response surface method in the reduced parameter space, and a sensitivity analysis based on such surrogate model. The few selected shapes were finally used to set up high fidelity RANS simulations and select an optimal shape.

@inproceedings{MolaTezzeleGadallaValdenazziGrassiPadovanRozza2019,
author = {Mola, Andrea and Tezzele, Marco and Gadalla, Mahmoud and Valdenazzi, Federica and Grassi, Davide and Padovan, Roberta and Rozza, Gianluigi},
title = {Efficient Reduction in Shape Parameter Space Dimension for Ship Propeller Blade Design},
year = {2019},
booktitle = {VIII International Conference on Computational Methods in Marine Engineering},
preprint = {https://arxiv.org/abs/1905.09815},
abstract = {In this work, we present the results of a ship propeller design optimization campaign carried out in the framework of the research project PRELICA, funded by the Friuli Venezia Giulia regional government. The main idea of this work is to operate on a multidisciplinary level to identify propeller shapes that lead to reduced tip vortex-induced pressure and increased efficiency without altering the thrust. First, a specific tool for the bottom-up construction of parameterized propeller blade geometries has been developed. The algorithm proposed operates with a user defined number of arbitrary shaped or NACA airfoil sections, and employs arbitrary degree NURBS to represent the chord, pitch, skew and rake distribution as a function of the blade radial coordinate. The control points of such curves have been modified to generate, in a fully automated way, a family of blade geometries depending on as many as 20 shape parameters. Such geometries have then been used to carry out potential flow simulations with the Boundary Element Method based software PROCAL. Given the high number of parameters considered, such a preliminary stage allowed for a fast evaluation of the performance of several hundreds of shapes. In addition, the data obtained from the potential flow simulation allowed for the application of a parameter space reduction methodology based on active subspaces (AS) property, which suggested that the main propeller performance indices are, at a first but rather accurate approximation, only depending on a single parameter which is a linear combination of all the original geometric ones. AS analysis has also been used to carry out a constrained optimization exploiting response surface method in the reduced parameter space, and a sensitivity analysis based on such surrogate model. The few selected shapes were finally used to set up high fidelity RANS simulations and select an optimal shape.}
}

12. M. Nonino, F. Ballarin, G. Rozza, and Y. Maday, "Overcoming slowly decaying Kolmogorov n-width by transport maps: application to model order reduction of fluid dynamics and fluid–structure interaction problems", 2019.
In this work we focus on reduced order modelling for problems for which the resulting reduced basis spaces show a slow decay of the Kolmogorov n-width, or, in practical calculations, its computational surrogate given by the magnitude of the eigenvalues returned by a proper orthogonal decomposition on the solution manifold. In particular, we employ an additional preprocessing during the offline phase of the reduced basis method, in order to obtain smaller reduced basis spaces. Such preprocessing is based on the composition of the snapshots with a transport map, that is a family of smooth and invertible mappings that map the physical domain of the problem into itself. Two test cases are considered: a fluid moving in a domain with deforming walls, and a fluid past a rotating cylinder. Comparison between the results of the novel offline stage and the standard one is presented.

@unpublished{NoninoBallarinRozzaMaday2019,
author = {Monica Nonino and Francesco Ballarin and Gianluigi Rozza and Yvon Maday},
title = {Overcoming slowly decaying Kolmogorov n-width by transport maps: application to model order reduction of fluid dynamics and fluid--structure interaction problems},
year = {2019},
preprint = {https://arxiv.org/abs/1911.06598},
abstract = {In this work we focus on reduced order modelling for problems for which the resulting reduced basis spaces show a slow decay of the Kolmogorov n-width, or, in practical calculations, its computational surrogate given by the magnitude of the eigenvalues returned by a proper orthogonal decomposition on the solution manifold. In particular, we employ an additional preprocessing during the offline phase of the reduced basis method, in order to obtain smaller reduced basis spaces. Such preprocessing is based on the composition of the snapshots with a transport map, that is a family of smooth and invertible mappings that map the physical domain of the problem into itself. Two test cases are considered: a fluid moving in a domain with deforming walls, and a fluid past a rotating cylinder. Comparison between the results of the novel offline stage and the standard one is presented.}
}

13. F. Pichi and G. Rozza, "Reduced basis approaches for parametrized bifurcation problems held by non-linear Von Kármán equations", , 81(1), pp. 112–135, 2019.
This work focuses on the computationally efficient detection of the buckling phenomena and bifurcation analysis of the parametric Von Kármán plate equations based on reduced order methods and spectral analysis. The computational complexity - due to the fourth order derivative terms, the non-linearity and the parameter dependence - provides an interesting benchmark to test the importance of the reduction strategies, during the construction of the bifurcation diagram by varying the parameter(s). To this end, together the state equations, we carry out also an analysis of the linearized eigenvalue problem, that allows us to better understand the physical behaviour near the bifurcation points, where we lose the uniqueness of solution. We test this automatic methodology also in the two parameter case, understanding the evolution of the first buckling mode.} journal = {Journal of Scientific Computing

@article{PichiRozza2019,
author = {Pichi, Federico and Rozza, Gianluigi},
title = {Reduced basis approaches for parametrized bifurcation problems held by non-linear Von Kármán equations},
year = {2019},
preprint = {https://arxiv.org/abs/1804.02014},
abstract = {This work focuses on the computationally efficient detection of the buckling phenomena and bifurcation analysis of the parametric Von Kármán plate equations based on reduced order methods and spectral analysis. The computational complexity - due to the fourth order derivative terms, the non-linearity and the parameter dependence - provides an interesting benchmark to test the importance of the reduction strategies, during the construction of the bifurcation diagram by varying the parameter(s). To this end, together the state equations, we carry out also an analysis of the linearized eigenvalue problem, that allows us to better understand the physical behaviour near the bifurcation points, where we lose the uniqueness of solution. We test this automatic methodology also in the two parameter case, understanding the evolution of the first buckling mode.}
journal = {Journal of Scientific Computing},
doi = {10.1007/s10915-019-01003-3},
volume = {81},
number = {1},
pages = {112--135},
}

14. M. Pintore, F. Pichi, M. Hess, G. Rozza, and C. Canuto, "Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method", 2019.
The majority of the most common physical phenomena can be described using partial differential equations (PDEs), however, they are very often characterized by strong nonlinearities. Such features lead to the coexistence of multiple solutions studied by the bifurcation theory. Unfortunately, in practical scenarios, one has to exploit numerical methods to compute the solutions of systems of PDEs, even if the classical techniques are usually able to compute only a single solution for any value of a parameter when more branches exist. In this work we implemented an elaborated deflated continuation method, that relies on the spectral element method (SEM) and on the reduced basis (RB) one, to efficiently compute bifurcation diagrams with more parameters and more bifurcation points. The deflated continuation method can be obtained combining the classical continuation method and the deflation one: the former is used to entirely track each known branch of the diagram, while the latter is exploited to discover the new ones. Finally, when more than one parameter is considered, the efficiency of the computation is ensured by the fact that the diagrams can be computed during the online phase while, during the offline one, one only has to compute one-dimensional diagrams. In this work, after a more detailed description of the method, we will show the results that can be obtained using it to compute a bifurcation diagram associated to a problem governed by the Navier-Stokes (NS) equations.

@unpublished{PintorePichiHessRozzaCanuto2019,
author = {Moreno Pintore and Federico Pichi and Martin Hess and Gianluigi Rozza and Claudio Canuto},
title = {Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method},
year = {2019},
preprint = {https://arxiv.org/abs/1912.06089},
abstract = {The majority of the most common physical phenomena can be described using partial differential equations (PDEs), however, they are very often characterized by strong nonlinearities. Such features lead to the coexistence of multiple solutions studied by the bifurcation theory. Unfortunately, in practical scenarios, one has to exploit numerical methods to compute the solutions of systems of PDEs, even if the classical techniques are usually able to compute only a single solution for any value of a parameter when more branches exist. In this work we implemented an elaborated deflated continuation method, that relies on the spectral element method (SEM) and on the reduced basis (RB) one, to efficiently compute bifurcation diagrams with more parameters and more bifurcation points. The deflated continuation method can be obtained combining the classical continuation method and the deflation one: the former is used to entirely track each known branch of the diagram, while the latter is exploited to discover the new ones. Finally, when more than one parameter is considered, the efficiency of the computation is ensured by the fact that the diagrams can be computed during the online phase while, during the offline one, one only has to compute one-dimensional diagrams. In this work, after a more detailed description of the method, we will show the results that can be obtained using it to compute a bifurcation diagram associated to a problem governed by the Navier-Stokes (NS) equations.}
}

15. N. V. Shah, M. Hess, and G. Rozza, "Discontinuous Galerkin Model Order Reduction of Geometrically Parametrized Stokes Equation", 2019.
The present work focuses on the geometric parametrization and the reduced order modeling of the Stokes equation. We discuss the concept of a parametrized geometry and its application within a reduced order modeling technique. The full order model is based on the discontinuous Galerkin method with an interior penalty formulation. We introduce the broken Sobolev spaces as well as the weak formulation required for an affine parameter dependency. The operators are transformed from a fixed domain to a parameter dependent domain using the affine parameter dependency. The proper orthogonal decomposition is used to obtain the basis of functions of the reduced order model. By using the Galerkin projection the linear system is projected onto the reduced space. During this process, the offline-online decomposition is used to separate parameter dependent operations from parameter independent operations. Finally this technique is applied to an obstacle test problem.The numerical outcomes presented include experimental error analysis, eigenvalue decay and measurement of online simulation time. Keywords: Discontinuous Galerkin method, Stokes flow, Geometric parametrization, Proper orthogonal decomposition.

@unpublished{ShahHessRozza2019,
author = {Nirav Vasant Shah and Martin Hess and Gianluigi Rozza},
title = {Discontinuous Galerkin Model Order Reduction of Geometrically Parametrized Stokes Equation},
year = {2019},
preprint = {https://arxiv.org/abs/1912.09787},
abstract = {The present work focuses on the geometric parametrization and the reduced order modeling of the Stokes equation. We discuss the concept of a parametrized geometry and its application within a reduced order modeling technique. The full order model is based on the discontinuous Galerkin method with an interior penalty formulation. We introduce the broken Sobolev spaces as well as the weak formulation required for an affine parameter dependency. The operators are transformed from a fixed domain to a parameter dependent domain using the affine parameter dependency. The proper orthogonal decomposition is used to obtain the basis of functions of the reduced order model. By using the Galerkin projection the linear system is projected onto the reduced space. During this process, the offline-online decomposition is used to separate parameter dependent operations from parameter independent operations. Finally this technique is applied to an obstacle test problem.The numerical outcomes presented include experimental error analysis, eigenvalue decay and measurement of online simulation time. Keywords: Discontinuous Galerkin method, Stokes flow, Geometric parametrization, Proper orthogonal decomposition.}
}

16. G. Stabile, F. Ballarin, G. Zuccarino, and G. Rozza, "A reduced order variational multiscale approach for turbulent flows", Advances in Computational Mathematics, 45(5), pp. 2349-2368, 2019.
The purpose of this work is to present a reduced order modeling framework for parametrized turbulent flows with moderately high Reynolds numbers within the variational multiscale (VMS) method. The Reduced Order Models (ROMs) presented in this manuscript are based on a POD-Galerkin approach with a VMS stabilization technique. Two different reduced order models are presented, which differ on the stabilization used during the Galerkin projection. In the first case the VMS stabilization method is used at both the full order and the reduced order level. In the second case, the VMS stabilization is used only at the full order level, while the projection of the standard Navier-Stokes equations is performed instead at the reduced order level. The former method is denoted as consistent ROM, while the latter is named non-consistent ROM, in order to underline the different choices made at the two levels. Particular attention is also devoted to the role of inf-sup stabilization by means of supremizers in ROMs based on a VMS formulation. Finally, the developed methods are tested on a numerical benchmark.

@article{StabileBallarinZuccarinoRozza2019,
author = {Stabile, Giovanni and Ballarin, Francesco and Zuccarino, Giacomo and Rozza, Gianluigi},
title = {A reduced order variational multiscale approach for turbulent flows},
year = {2019},
journal = {Advances in Computational Mathematics},
volume = {45},
number = {5},
pages = {2349-2368},
doi = {10.1007/s10444-019-09712-x},
preprint = {https://arxiv.org/abs/1809.11101},
abstract = {The purpose of this work is to present a reduced order modeling framework for parametrized turbulent flows with moderately high Reynolds numbers within the variational multiscale (VMS) method. The Reduced Order Models (ROMs) presented in this manuscript are based on a POD-Galerkin approach with a VMS stabilization technique. Two different reduced order models are presented, which differ on the stabilization used during the Galerkin projection. In the first case the VMS stabilization method is used at both the full order and the reduced order level. In the second case, the VMS stabilization is used only at the full order level, while the projection of the standard Navier-Stokes equations is performed instead at the reduced order level. The former method is denoted as consistent ROM, while the latter is named non-consistent ROM, in order to underline the different choices made at the two levels. Particular attention is also devoted to the role of inf-sup stabilization by means of supremizers in ROMs based on a VMS formulation. Finally, the developed methods are tested on a numerical benchmark.}
}

17. K. S. Star, G. Stabile, F. Belloni, G. Rozza, and J. Degroote, "Extension and comparison of techniques to enforce boundary conditions in Finite Volume POD-Galerkin reduced order models for fluid dynamic problems", 2019.
A Finite-Volume based POD-Galerkin reduced order model is developed for fluid dynamic problems where the (time-dependent) boundary conditions are controlled using two different boundary control strategies: the control function method, whose aim is to obtain homogeneous basis functions for the reduced basis space and the penalty method where the boundary conditions are enforced in the reduced order model using a penalty factor. The penalty method is improved by using an iterative solver for the determination of the penalty factor rather than tuning the factor with a sensitivity analysis or numerical experimentation. The boundary control methods are compared and tested for two cases: the classical lid driven cavity benchmark problem and a Y-junction flow case with two inlet channels and one outlet channel. The results show that the boundaries of the reduced order model can be controlled with the boundary control methods and the same order of accuracy is achieved for the velocity and pressure fields. Finally, the speedup ratio between the reduced order models and the full order model is of the order 1000 for the lid driven cavity case and of the order 100 for the Y-junction test case.

@unpublished{StarStabileBelloniRozzaDegroote2019,
author = {S. Kelbij Star and Giovanni Stabile and Francesco Belloni and Gianluigi Rozza and Joris Degroote},
title = {Extension and comparison of techniques to enforce boundary conditions in Finite Volume POD-Galerkin reduced order models for fluid dynamic problems},
year = {2019},
preprint = {https://arxiv.org/abs/1912.00825},
abstract = {A Finite-Volume based POD-Galerkin reduced order model is developed for fluid dynamic problems where the (time-dependent) boundary conditions are controlled using two different boundary control strategies: the control function method, whose aim is to obtain homogeneous basis functions for the reduced basis space and the penalty method where the boundary conditions are enforced in the reduced order model using a penalty factor. The penalty method is improved by using an iterative solver for the determination of the penalty factor rather than tuning the factor with a sensitivity analysis or numerical experimentation. The boundary control methods are compared and tested for two cases: the classical lid driven cavity benchmark problem and a Y-junction flow case with two inlet channels and one outlet channel. The results show that the boundaries of the reduced order model can be controlled with the boundary control methods and the same order of accuracy is achieved for the velocity and pressure fields. Finally, the speedup ratio between the reduced order models and the full order model is of the order 1000 for the lid driven cavity case and of the order 100 for the Y-junction test case.}
}

18. M. Tezzele, N. Demo, and G. Rozza, "Shape optimization through proper orthogonal decomposition with interpolation and dynamic mode decomposition enhanced by active subspaces", in VIII International Conference on Computational Methods in Marine Engineering, 2019.
We propose a numerical pipeline for shape optimization in naval engineering involving two different non-intrusive reduced order method (ROM) techniques. Such methods are proper orthogonal decomposition with interpolation (PODI) and dynamic mode decomposition (DMD). The ROM proposed will be enhanced by active subspaces (AS) as a pre-processing tool that reduce the parameter space dimension and suggest better sampling of the input space. We will focus on geometrical parameters describing the perturbation of a reference bulbous bow through the free form deformation (FFD) technique. The ROM are based on a finite volume method (FV) to simulate the multi-phase incompressible flow around the deformed hulls. In previous works we studied the reduction of the parameter space in naval engineering through AS [38, 10] focusing on different parts of the hull. PODI and DMD have been employed for the study of fast and reliable shape optimization cycles on a bulbous bow in [9]. The novelty of this work is the simultaneous reduction of both the input parameter space and the output fields of interest. In particular AS will be trained computing the total drag resistance of a hull advancing in calm water and its gradients with respect to the input parameters. DMD will improve the performance of each simulation of the campaign using only few snapshots of the solution fields in order to predict the regime state of the system. Finally PODI will interpolate the coefficients of the POD decomposition of the output fields for a fast approximation of all the fields at new untried parameters given by the optimization algorithm. This will result in a non-intrusive data-driven numerical optimization pipeline completely independent with respect to the full order solver used and it can be easily incorporated into existing numerical pipelines, from the reference CAD to the optimal shape.

@inproceedings{TezzeleDemoRozza2019,
author = {Tezzele, Marco and Demo, Nicola and Rozza, Gianluigi},
title = {Shape optimization through proper orthogonal decomposition with interpolation and dynamic mode decomposition enhanced by active subspaces},
booktitle = {VIII International Conference on Computational Methods in Marine Engineering},
year = {2019},
preprint = {https://arxiv.org/abs/1905.05483},
abstract = {We propose a numerical pipeline for shape optimization in naval engineering involving two different non-intrusive reduced order method (ROM) techniques. Such methods are proper orthogonal decomposition with interpolation (PODI) and dynamic mode decomposition (DMD). The ROM proposed will be enhanced by active subspaces (AS) as a pre-processing tool that reduce the parameter space dimension and suggest better sampling of the input space. We will focus on geometrical parameters describing the perturbation of a reference bulbous bow through the free form deformation (FFD) technique. The ROM are based on a finite volume method (FV) to simulate the multi-phase incompressible flow around the deformed hulls. In previous works we studied the reduction of the parameter space in naval engineering through AS [38, 10] focusing on different parts of the hull. PODI and DMD have been employed for the study of fast and reliable shape optimization cycles on a bulbous bow in [9]. The novelty of this work is the simultaneous reduction of both the input parameter space and the output fields of interest. In particular AS will be trained computing the total drag resistance of a hull advancing in calm water and its gradients with respect to the input parameters. DMD will improve the performance of each simulation of the campaign using only few snapshots of the solution fields in order to predict the regime state of the system. Finally PODI will interpolate the coefficients of the POD decomposition of the output fields for a fast approximation of all the fields at new untried parameters given by the optimization algorithm. This will result in a non-intrusive data-driven numerical optimization pipeline completely independent with respect to the full order solver used and it can be easily incorporated into existing numerical pipelines, from the reference CAD to the optimal shape.}
}

19. L. Venturi, D. Torlo, F. Ballarin, and G. Rozza, "Weighted Reduced Order Methods for Parametrized Partial Differential Equations with Random Inputs", in Uncertainty Modeling for Engineering Applications, F. Canavero (ed.), Springer International Publishing, pp. 27–40, 2019.
In this manuscript we discuss weighted reduced order methods for stochastic partial differential equations. Random inputs (such as forcing terms, equation coefficients, boundary conditions) are considered as parameters of the equations. We take advantage of the resulting parametrized formulation to propose an efficient reduced order model; we also profit by the underlying stochastic assumption in the definition of suitable weights to drive to reduction process. Two viable strategies are discussed, namely the weighted reduced basis method and the weighted proper orthogonal decomposition method. A numerical example on a parametrized elasticity problem is shown.

@inbook{VenturiTorloBallarinRozza2019,
author = {Venturi, Luca and Torlo, Davide and Ballarin, Francesco and Rozza, Gianluigi},
chapter = {Weighted Reduced Order Methods for Parametrized Partial Differential Equations with Random Inputs},
year = {2019},
booktitle = {Uncertainty Modeling for Engineering Applications},
editor = {Canavero, Flavio},
publisher = {Springer International Publishing},
pages = {27--40},
preprint = {https://arxiv.org/abs/1805.00828},
doi = {10.1007/978-3-030-04870-9_2},
abstract = {In this manuscript we discuss weighted reduced order methods for stochastic partial differential equations. Random inputs (such as forcing terms, equation coefficients, boundary conditions) are considered as parameters of the equations. We take advantage of the resulting parametrized formulation to propose an efficient reduced order model; we also profit by the underlying stochastic assumption in the definition of suitable weights to drive to reduction process. Two viable strategies are discussed, namely the weighted reduced basis method and the weighted proper orthogonal decomposition method. A numerical example on a parametrized elasticity problem is shown.}
}

20. M. Zancanaro, F. Ballarin, S. Perotto, and G. Rozza, "Hierarchical Model Reduction Techniques for Flow Modeling in a Parametrized Setting", 2019.
In this work we focus on two different methods to deal with parametrized partial differential equations in an efficient and accurate way. Starting from high fidelity approximations built via the hierarchical model reduction discretization, we consider two approaches, both based on a projection model reduction technique. The two methods differ for the algorithm employed during the construction of the reduced basis. In particular, the former employs the proper orthogonal decomposition, while the latter relies on a greedy algorithm according to the certified reduced basis technique. The two approaches are preliminarily compared on two-dimensional scalar and vector test cases.

@unpublished{ZancanaroBallarinPerottoRozza2019,
author = {Matteo Zancanaro and Francesco Ballarin and Simona Perotto and Gianluigi Rozza},
title = {Hierarchical Model Reduction Techniques for Flow Modeling in a Parametrized Setting},
year = {2019},
preprint = {https://arxiv.org/abs/1909.01668},
abstract = {In this work we focus on two different methods to deal with parametrized partial differential equations in an efficient and accurate way. Starting from high fidelity approximations built via the hierarchical model reduction discretization, we consider two approaches, both based on a projection model reduction technique. The two methods differ for the algorithm employed during the construction of the reduced basis. In particular, the former employs the proper orthogonal decomposition, while the latter relies on a greedy algorithm according to the certified reduced basis technique. The two approaches are preliminarily compared on two-dimensional scalar and vector test cases.}
}

### 2018

1. F. Auricchio, M. Conti, A. Lefieux, S. Morganti, A. Reali, G. Rozza, and A. Veneziani, "Computational methods in cardiovascular mechanics", in Cardiovascular Mechanics, M. F. Labrosse (ed.), CRC Press, pp. 54, 2018.
The introduction of computational models in cardiovascular sciences has been progressively bringing new and unique tools for the investigation of the physiopathology. Together with the dramatic improvement of imaging and measuring devices on one side, and of computational architectures on the other one, mathematical and numerical models have provided a new, clearly noninvasive, approach for understanding not only basic mechanisms but also patient-specific conditions, and for supporting the design and the development of new therapeutic options. The terminology in silico is, nowadays, commonly accepted for indicating this new source of knowledge added to traditional in vitro and in vivo investigations. The advantages of in silico methodologies are basically the low cost in terms of infrastructures and facilities, the reduced invasiveness and, in general, the intrinsic predictive capabilities based on the use of mathematical models. The disadvantages are generally identified in the distance between the real cases and their virtual counterpart required by the conceptual modeling that can be detrimental for the reliability of numerical simulations.

@inbook{AuricchioContiLefieuxMorgantiRealiRozzaVeneziani2018,
author = {Auricchio, Ferdinando and Conti, Michele and Lefieux, Adrian and Morganti, Simone and Reali, Alessandro and Rozza, Gianluigi and Veneziani, Alessandro},
editor = {Michel F. Labrosse},
booktitle = {Cardiovascular Mechanics},
chapter = {Computational methods in cardiovascular mechanics},
year = {2018},
pages = {54},
publisher = {CRC Press},
preprint = {https://arxiv.org/abs/1803.04535},
url = {https://www.taylorfrancis.com/books/e/9781315280288/chapters/10.1201%2Fb21917-5},
abstract = {The introduction of computational models in cardiovascular sciences has been progressively bringing new and unique tools for the investigation of the physiopathology. Together with the dramatic improvement of imaging and measuring devices on one side, and of computational architectures on the other one, mathematical and numerical models have provided a new, clearly noninvasive, approach for understanding not only basic mechanisms but also patient-specific conditions, and for supporting the design and the development of new therapeutic options. The terminology in silico is, nowadays, commonly accepted for indicating this new source of knowledge added to traditional in vitro and in vivo investigations. The advantages of in silico methodologies are basically the low cost in terms of infrastructures and facilities, the reduced invasiveness and, in general, the intrinsic predictive capabilities based on the use of mathematical models. The disadvantages are generally identified in the distance between the real cases and their virtual counterpart required by the conceptual modeling that can be detrimental for the reliability of numerical simulations.}
}

2. F. Ballarin, A. D'Amario, S. Perotto, and G. Rozza, "A POD-selective inverse distance weighting method for fast parametrized shape morphing", International Journal for Numerical Methods in Engineering, 117(8), pp. 860–884, 2018.
Efficient shape morphing techniques play a crucial role in the approximation of partial differential equations defined in parametrized domains, such as for fluid-structure interaction or shape optimization problems. In this paper, we focus on Inverse Distance Weighting (IDW) interpolation techniques, where a reference domain is morphed into a deformed one via the displacement of a set of control points. We aim at reducing the computational burden characterizing a standard IDW approach without compromising the accuracy. To this aim, first we propose an improvement of IDW based on a geometric criterion which automatically selects a subset of the original set of control points. Then, we combine this new approach with a model reduction technique based on a Proper Orthogonal Decomposition of the set of admissible displacements. This choice further reduces computational costs. We verify the performances of the new IDW techniques on several tests by investigating the trade-off reached in terms of accuracy and efficiency.

@article{BallarinDAmarioPerottoRozza2018,
author = {Ballarin, Francesco and D'Amario, Alessandro and Perotto, Simona and Rozza, Gianluigi},
title = {A POD-selective inverse distance weighting method for fast parametrized shape morphing},
year = {2018},
preprint = {https://arxiv.org/abs/1710.09243},
doi = {10.1002/nme.5982},
volume = {117},
number = {8},
pages = {860--884},
journal = {International Journal for Numerical Methods in Engineering},
abstract = {Efficient shape morphing techniques play a crucial role in the approximation of partial differential equations defined in parametrized domains, such as for fluid-structure interaction or shape optimization problems. In this paper, we focus on Inverse Distance Weighting (IDW) interpolation techniques, where a reference domain is morphed into a deformed one via the displacement of a set of control points. We aim at reducing the computational burden characterizing a standard IDW approach without compromising the accuracy. To this aim, first we propose an improvement of IDW based on a geometric criterion which automatically selects a subset of the original set of control points. Then, we combine this new approach with a model reduction technique based on a Proper Orthogonal Decomposition of the set of admissible displacements. This choice further reduces computational costs. We verify the performances of the new IDW techniques on several tests by investigating the trade-off reached in terms of accuracy and efficiency.}
}

3. D. Boffi, L. F. Pavarino, G. Rozza, S. Scacchi, and C. Vergara, Mathematical and Numerical Modeling of the Cardiovascular System and Applications), Springer International Publishing, 2018.
[BibTeX] [View on publisher website]
@book{BoffiPavarinoRozzaScachiVergara2018,
doi = {10.1007/978-3-319-96649-6},
year = {2018},
publisher = {Springer International Publishing},
author = {Daniele Boffi and Luca F. Pavarino and Gianluigi Rozza and Simone Scacchi and Christian Vergara},
title = {Mathematical and Numerical Modeling of the Cardiovascular System and Applications}
}

4. N. Demo, M. Tezzele, and G. Rozza, "EZyRB: Easy Reduced Basis method", Journal of Open Source Software, 3(24), pp. 661, 2018.
@article{DemoTezzeleRozza2018b,
author = {Demo, Nicola and Tezzele, Marco and Rozza, Gianluigi},
title = {EZyRB: Easy Reduced Basis method},
year = {2018},
preprint = {https://www.theoj.org/joss-papers/joss.00661/10.21105.joss.00661.pdf},
journal = {Journal of Open Source Software},
doi = {10.21105/joss.00661},
volume = {3},
number = {24},
pages = {661}
}

5. N. Demo, M. Tezzele, and G. Rozza, "PyDMD: Python Dynamic Mode Decomposition", Journal of Open Source Software, 3(22), pp. 530, 2018.
@article{DemoTezzeleRozza2018a,
author = {Demo, Nicola and Tezzele, Marco and Rozza, Gianluigi},
title = {PyDMD: Python Dynamic Mode Decomposition},
year = {2018},
preprint = {https://www.theoj.org/joss-papers/joss.00530/10.21105.joss.00530.pdf},
journal = {Journal of Open Source Software},
doi = {10.21105/joss.00530},
volume = {3},
number = {22},
pages = {530}
}

6. N. Demo, M. Tezzele, G. Gustin, G. Lavini, and G. Rozza, "Shape Optimization by means of Proper Orthogonal Decomposition and Dynamic Mode Decomposition", in Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research, 2018, pp. 212–219.
Shape optimization is a challenging task in many engineering fields, since the numerical solutions of parametric system may be computationally expensive. This work presents a novel optimization procedure based on reduced order modeling, applied to a naval hull design problem. The advantage introduced by this method is that the solution for a specific parameter can be expressed as the combination of few numerical solutions computed at properly chosen parametric points. The reduced model is built using the proper orthogonal decomposition with interpolation (PODI) method. We use the free form deformation (FFD) for an automated perturbation of the shape, and the finite volume method to simulate the multiphase incompressible flow around the deformed hulls. Further computational reduction is done by the dynamic mode decomposition (DMD) technique: from few high dimensional snapshots, the system evolution is reconstructed and the final state of the simulation is faithfully approximated. Finally the global optimization algorithm iterates over the reduced space: the approximated drag and lift coefficients are projected to the hull surface, hence the resistance is evaluated for the new hulls until the convergence to the optimal shape is achieved. We will present the results obtained applying the described procedure to a typical Fincantieri cruise ship

@inproceedings{DemoTezzeleGustinLaviniRozza2018,
author = {Demo, Nicola and Tezzele, Marco and Gustin, Gianluca and Lavini, Gianpiero and Rozza, Gianluigi},
booktitle = {Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship \& Maritime Research},
title = {Shape Optimization by means of Proper Orthogonal Decomposition and Dynamic Mode Decomposition},
year = {2018},
preprint = {https://arxiv.org/abs/1803.07368},
doi = {10.3233/978-1-61499-870-9-212},
pages = {212--219},
publisher = {IOS Press},
abstract = {Shape optimization is a challenging task in many engineering fields, since the numerical solutions of parametric system may be computationally expensive. This work presents a novel optimization procedure based on reduced order modeling, applied to a naval hull design problem. The advantage introduced by this method is that the solution for a specific parameter can be expressed as the combination of few numerical solutions computed at properly chosen parametric points. The reduced model is built using the proper orthogonal decomposition with interpolation (PODI) method. We use the free form deformation (FFD) for an automated perturbation of the shape, and the finite volume method to simulate the multiphase incompressible flow around the deformed hulls. Further computational reduction is done by the dynamic mode decomposition (DMD) technique: from few high dimensional snapshots, the system evolution is reconstructed and the final state of the simulation is faithfully approximated. Finally the global optimization algorithm iterates over the reduced space: the approximated drag and lift coefficients are projected to the hull surface, hence the resistance is evaluated for the new hulls until the convergence to the optimal shape is achieved. We will present the results obtained applying the described procedure to a typical Fincantieri cruise ship}
}

7. N. Demo, M. Tezzele, A. Mola, and G. Rozza, "An efficient shape parametrisation by free-form deformation enhanced by active subspace for hull hydrodynamic ship design problems in open source environment", in The 28th International Ocean and Polar Engineering Conference, 2018.
In this contribution, we present the results of the application of a parameter space reduction methodology based on active subspaces to the hull hydrodynamic design problem. Several parametric deformations of an initial hull shape are considered to assess the influence of the shape parameters considered on the hull total drag. The hull resistance is typically computed by means of numerical simulations of the hydrodynamic flow past the ship. Given the high number of parameters involved - which might result in a high number of time consuming hydrodynamic simulations - assessing whether the parameters space can be reduced would lead to considerable computational cost reduction. Thus, the main idea of this work is to employ the active subspaces to identify possible lower dimensional structures in the parameter space, or to verify the parameter distribution in the position of the control points. To this end, a fully automated procedure has been implemented to produce several small shape perturbations of an original hull CAD geometry which are then used to carry out high-fidelity flow simulations and collect data for the active subspaces analysis. To achieve full automation of the open source pipeline described, both the free form deformation methodology employed for the hull perturbations and the solver based on unsteady potential flow theory, with fully nonlinear free surface treatment, are directly interfaced with CAD data structures and operate using IGES vendor-neutral file formats as input files. The computational cost of the fluid dynamic simulations is further reduced through the application of dynamic mode decomposition to reconstruct the steady state total drag value given only few initial snapshots of the simulation. The active subspaces analysis is here applied to the geometry of the DTMB-5415 naval combatant hull, which is a common benchmark in ship hydrodynamics simulations.

@inproceedings{DemoTezzeleMolaRozza2018,
author = {Demo, Nicola and Tezzele, Marco and Mola, Andrea and Rozza, Gianluigi},
title = {An efficient shape parametrisation by free-form deformation enhanced by active subspace for hull hydrodynamic ship design problems in open source environment},
booktitle = {The 28th International Ocean and Polar Engineering Conference},
year = {2018},
preprint = {https://arxiv.org/abs/1801.06369},
abstract = {In this contribution, we present the results of the application of a parameter space reduction methodology based on active subspaces to the hull hydrodynamic design problem. Several parametric deformations of an initial hull shape are considered to assess the influence of the shape parameters considered on the hull total drag. The hull resistance is typically computed by means of numerical simulations of the hydrodynamic flow past the ship. Given the high number of parameters involved - which might result in a high number of time consuming hydrodynamic simulations - assessing whether the parameters space can be reduced would lead to considerable computational cost reduction. Thus, the main idea of this work is to employ the active subspaces to identify possible lower dimensional structures in the parameter space, or to verify the parameter distribution in the position of the control points. To this end, a fully automated procedure has been implemented to produce several small shape perturbations of an original hull CAD geometry which are then used to carry out high-fidelity flow simulations and collect data for the active subspaces analysis. To achieve full automation of the open source pipeline described, both the free form deformation methodology employed for the hull perturbations and the solver based on unsteady potential flow theory, with fully nonlinear free surface treatment, are directly interfaced with CAD data structures and operate using IGES vendor-neutral file formats as input files. The computational cost of the fluid dynamic simulations is further reduced through the application of dynamic mode decomposition to reconstruct the steady state total drag value given only few initial snapshots of the simulation. The active subspaces analysis is here applied to the geometry of the DTMB-5415 naval combatant hull, which is a common benchmark in ship hydrodynamics simulations.}
}

8. F. Garotta, N. Demo, M. Tezzele, M. Carraturo, A. Reali, and G. Rozza, "Reduced Order Isogeometric Analysis Approach for PDEs in Parametrized Domains", 2018.
In this contribution, we coupled the isogeometric analysis to a reduced order modelling technique in order to provide a computationally efficient solution in parametric domains. In details, we adopt the free-form deformation method to obtain the parametric formulation of the domain and proper orthogonal decomposition with interpolation for the computational reduction of the model. This technique provides a real-time solution for any parameter by combining several solutions, in this case computed using isogeometric analysis on different geometrical configurations of the domain, properly mapped into a reference configuration. We underline that this reduced order model requires only the full-order solutions, making this approach non-intrusive. We present in this work the results of the application of this methodology to a heat conduction problem inside a deformable collector pipe.

@unpublished{GarottaDemoTezzeleCaraturroRealiRozza2018,
author = {Garotta, Fabrizio and Demo, Nicola and Tezzele, Marco and Carraturo, Massimo and Reali, Alessandro and Rozza, Gianluigi},
title = {Reduced Order Isogeometric Analysis Approach for PDEs in Parametrized Domains},
year = {2018},
preprint = {https://arxiv.org/abs/1811.08631},
abstract = {In this contribution, we coupled the isogeometric analysis to a reduced order modelling technique in order to provide a computationally efficient solution in parametric domains. In details, we adopt the free-form deformation method to obtain the parametric formulation of the domain and proper orthogonal decomposition with interpolation for the computational reduction of the model. This technique provides a real-time solution for any parameter by combining several solutions, in this case computed using isogeometric analysis on different geometrical configurations of the domain, properly mapped into a reference configuration. We underline that this reduced order model requires only the full-order solutions, making this approach non-intrusive. We present in this work the results of the application of this methodology to a heat conduction problem inside a deformable collector pipe.}
}

9. D. B. P. Huynh, F. Pichi, and G. Rozza, "Reduced basis approximation and a posteriori error estimation: applications to elasticity problems in several parametric settings", in Numerical Methods for PDEs: State of the Art Techniques, D. A. Di Pietro, A. Ern, and L. Formaggia (eds.), Springer International Publishing, vol. 15, pp. 203–247, 2018.
In this work we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for elasticity problems in affinley parametrized geometries. The essential ingredients of the methodology are: a Galerkin projection onto a low-dimensional space associated with a smooth "parametric manifold" - dimension reduction, an efficient and effective greedy sampling methods for identification of optimal and numerically stable approximations - rapid convergence, an a posteriori error estimation procedures - rigorous and sharp bounds for the functional outputs related with the underlying solution or related quantities of interest, like stress intensity factor, and Offline-Online computational decomposition strategies - minimum marginal cost for high performance in the real-time and many-query (e.g., design and optimization) contexts. We present several illustrative results for linear elasticity problem in parametrized geometries representing 2D Cartesian or 3D axisymmetric configurations like an arc-cantilever beam, a center crack problem, a composite unit cell or a woven composite beam, a multi-material plate, and a closed vessel. We consider different parametrization for the systems: either physical quantities - to model the materials and loads - and geometrical parameters - to model different geometrical configurations - with isotropic and orthotropic materials working in plane stress and plane strain approximation. We would like to underline the versatility of the methodology in very different problems. As last example we provide a nonlinear setting with increased complexity.

@inbook{HuynhPichiRozza2018,
author = {Huynh, Dinh Bao Phuong and Pichi, Federico and Rozza, Gianluigi},
chapter = {Reduced basis approximation and a posteriori error estimation: applications to elasticity problems in several parametric settings},
year = {2018},
editor = {Di Pietro, Daniele Antonio and Ern, Alexandre and Formaggia, Luca},
booktitle = {Numerical Methods for PDEs: State of the Art Techniques},
publisher = {Springer International Publishing},
pages = {203--247},
volume = {15},
doi = {10.1007/978-3-319-94676-4_8},
preprint = {https://arxiv.org/abs/1801.06553},
abstract = {In this work we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for elasticity problems in affinley parametrized geometries. The essential ingredients of the methodology are: a Galerkin projection onto a low-dimensional space associated with a smooth "parametric manifold" - dimension reduction, an efficient and effective greedy sampling methods for identification of optimal and numerically stable approximations - rapid convergence, an a posteriori error estimation procedures - rigorous and sharp bounds for the functional outputs related with the underlying solution or related quantities of interest, like stress intensity factor, and Offline-Online computational decomposition strategies - minimum marginal cost for high performance in the real-time and many-query (e.g., design and optimization) contexts. We present several illustrative results for linear elasticity problem in parametrized geometries representing 2D Cartesian or 3D axisymmetric configurations like an arc-cantilever beam, a center crack problem, a composite unit cell or a woven composite beam, a multi-material plate, and a closed vessel. We consider different parametrization for the systems: either physical quantities - to model the materials and loads - and geometrical parameters - to model different geometrical configurations - with isotropic and orthotropic materials working in plane stress and plane strain approximation. We would like to underline the versatility of the methodology in very different problems. As last example we provide a nonlinear setting with increased complexity.}
}

10. I. Martini, B. Haasdonk, and G. Rozza, "Certified Reduced Basis Approximation for the Coupling of Viscous and Inviscid Parametrized Flow Models", Journal of Scientific Computing, 74, pp. 197-219, 2018.
[BibTeX] [View on publisher website]
@article{MartiniHaasdonkRozza2018,
title = {Certified Reduced Basis Approximation for the Coupling of Viscous and Inviscid Parametrized Flow Models},
journal = {Journal of Scientific Computing},
volume = {74},
year = {2018},
pages = {197-219},
doi = {10.1007/s10915-017-0430-y},
author = {Martini, I. and Haasdonk, B. and Rozza, G.}
}

11. G. Rozza, H. Malik, N. Demo, M. Tezzele, M. Girfoglio, G. Stabile, and A. Mola, "Advances in Reduced Order Methods for Parametric Industrial Problems in Computational Fluid Dynamics", 2018.
Reduced order modeling has gained considerable attention in recent decades owing to the advantages offered in reduced computational times and multiple solutions for parametric problems. The focus of this manuscript is the application of model order reduction techniques in various engineering and scientific applications including but not limited to mechanical, naval and aeronautical engineering. The focus here is kept limited to computational fluid mechanics and related applications. The advances in the reduced order modeling with proper orthogonal decomposition and reduced basis method are presented as well as a brief discussion of dynamic mode decomposition and also some present advances in the parameter space reduction. Here, an overview of the challenges faced and possible solutions are presented with examples from various problems.

@unpublished{RozzaMalikDemoTezzeleGirfoglioStabileMola2018,
author = {Rozza, Gianluigi and Malik, Haris and Demo, Nicola and Tezzele, Marco and Girfoglio, Michele and Stabile, Giovanni and Mola, Andrea},
title = {Advances in Reduced Order Methods for Parametric Industrial Problems in Computational Fluid Dynamics},
year = {2018},
preprint = {https://arxiv.org/abs/1811.08319},
abstract = {Reduced order modeling has gained considerable attention in recent decades owing to the advantages offered in reduced computational times and multiple solutions for parametric problems. The focus of this manuscript is the application of model order reduction techniques in various engineering and scientific applications including but not limited to mechanical, naval and aeronautical engineering. The focus here is kept limited to computational fluid mechanics and related applications. The advances in the reduced order modeling with proper orthogonal decomposition and reduced basis method are presented as well as a brief discussion of dynamic mode decomposition and also some present advances in the parameter space reduction. Here, an overview of the challenges faced and possible solutions are presented with examples from various problems.}
}

12. F. Salmoiraghi, A. Scardigli, H. Telib, and G. Rozza, "Free-form deformation, mesh morphing and reduced-order methods: enablers for efficient aerodynamic shape optimisation", International Journal of Computational Fluid Dynamics, 32(4-5), pp. 233-247, 2018.
In this work, we provide an integrated pipeline for the model-order reduction of turbulent flows around parametrised geometries in aerodynamics. In particular, free-form deformation is applied for geometry parametrisation, whereas two different reduced-order models based on proper orthogonal decomposition (POD) are employed in order to speed-up the full-order simulations: the first method exploits POD with interpolation, while the second one is based on domain decomposition. For the sampling of the parameter space, we adopt a Greedy strategy coupled with Constrained Centroidal Voronoi Tessellations, in order to guarantee a good compromise between space exploration and exploitation. The proposed framework is tested on an industrially relevant application, i.e. the front-bumper morphing of the DrivAer car model, using the finite-volume method for the full-order resolution of the Reynolds-Averaged Navier–Stokes equations.

@article{SalmoiraghiScardigliTelibRozza2018,
author = {Salmoiraghi, Filippo and Scardigli, Angela and Telib, Haysam and Rozza, Gianluigi},
title = {Free-form deformation, mesh morphing and reduced-order methods: enablers for efficient aerodynamic shape optimisation},
year = {2018},
preprint = {https://arxiv.org/abs/1803.04688},
abstract = {In this work, we provide an integrated pipeline for the model-order reduction of turbulent flows around parametrised geometries in aerodynamics. In particular, free-form deformation is applied for geometry parametrisation, whereas two different reduced-order models based on proper orthogonal decomposition (POD) are employed in order to speed-up the full-order simulations: the first method exploits POD with interpolation, while the second one is based on domain decomposition. For the sampling of the parameter space, we adopt a Greedy strategy coupled with Constrained Centroidal Voronoi Tessellations, in order to guarantee a good compromise between space exploration and exploitation. The proposed framework is tested on an industrially relevant application, i.e. the front-bumper morphing of the DrivAer car model, using the finite-volume method for the full-order resolution of the Reynolds-Averaged Navier–Stokes equations.},
journal = {International Journal of Computational Fluid Dynamics},
volume = {32},
number = {4-5},
pages = {233-247},
year = {2018},
publisher = {Taylor & Francis},
doi = {10.1080/10618562.2018.1514115}
}

13. G. Stabile and G. Rozza, "Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier–Stokes equations", Computers & Fluids, 173, pp. 273–284, 2018.
In this work a stabilised and reduced Galerkin projection of the incompressible unsteady Navier–Stokes equations for moderate Reynolds number is presented. The full-order model, on which the Galerkin projection is applied, is based on a finite volumes approximation. The reduced basis spaces are constructed with a POD approach. Two different pressure stabilisation strategies are proposed and compared: the former one is based on the supremizer enrichment of the velocity space, and the latter one is based on a pressure Poisson equation approach.

@article{StabileRozza2018,
title = {Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier--Stokes equations},
journal = {Computers & Fluids},
year = {2018},
abstract = {In this work a stabilised and reduced Galerkin projection of the incompressible unsteady Navier–Stokes equations for moderate Reynolds number is presented. The full-order model, on which the Galerkin projection is applied, is based on a finite volumes approximation. The reduced basis spaces are constructed with a POD approach. Two different pressure stabilisation strategies are proposed and compared: the former one is based on the supremizer enrichment of the velocity space, and the latter one is based on a pressure Poisson equation approach.},
preprint = {https://arxiv.org/abs/1710.11580},
author = {Stabile, Giovanni and Gianluigi Rozza},
doi = {10.1016/j.compfluid.2018.01.035},
volume = {173},
pages = {273--284},
}

14. M. Strazzullo, F. Ballarin, R. Mosetti, and G. Rozza, "Model Reduction for Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering", SIAM Journal on Scientific Computing, 40(4), pp. B1055-B1079, 2018.
We propose reduced order methods as a suitable approach to face parametrized optimal control problems governed by partial differential equations, with applications in en- vironmental marine sciences and engineering. Environmental parametrized optimal control problems are usually studied for different configurations described by several physical and/or geometrical parameters representing different phenomena and structures. The solution of parametrized problems requires a demanding computational effort. In order to save com- putational time, we rely on reduced basis techniques as a reliable and rapid tool to solve parametrized problems. We introduce general parametrized linear quadratic optimal control problems, and the saddle-point structure of their optimality system. Then, we propose a POD-Galerkin reduction of the optimality system. Finally, we test the resulting method on two environmental applications: a pollutant control in the Gulf of Trieste, Italy and a solution tracking governed by quasi-geostrophic equations describing North Atlantic Ocean dynamic. The two experiments underline how reduced order methods are a reliable and convenient tool to manage several environmental optimal control problems, for different mathematical models, geographical scale as well as physical meaning.

@article{StrazzulloBallarinMosettiRozza2018,
author = {Strazzullo, Maria and Ballarin, Francesco and Mosetti, Renzo and Rozza, Gianluigi},
title = {Model Reduction for Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering},
journal = {SIAM Journal on Scientific Computing},
volume = {40},
number = {4},
pages = {B1055-B1079},
year = {2018},
preprint = {https://arxiv.org/abs/1710.01640},
doi = {10.1137/17M1150591},
abstract = {We propose reduced order methods as a suitable approach to face parametrized optimal control problems governed by partial differential equations, with applications in en- vironmental marine sciences and engineering. Environmental parametrized optimal control problems are usually studied for different configurations described by several physical and/or geometrical parameters representing different phenomena and structures. The solution of parametrized problems requires a demanding computational effort. In order to save com- putational time, we rely on reduced basis techniques as a reliable and rapid tool to solve parametrized problems. We introduce general parametrized linear quadratic optimal control problems, and the saddle-point structure of their optimality system. Then, we propose a POD-Galerkin reduction of the optimality system. Finally, we test the resulting method on two environmental applications: a pollutant control in the Gulf of Trieste, Italy and a solution tracking governed by quasi-geostrophic equations describing North Atlantic Ocean dynamic. The two experiments underline how reduced order methods are a reliable and convenient tool to manage several environmental optimal control problems, for different mathematical models, geographical scale as well as physical meaning.}
}

15. M. Tezzele, N. Demo, M. Gadalla, A. Mola, and G. Rozza, "Model order reduction by means of active subspaces and dynamic mode decomposition for parametric hull shape design hydrodynamics", in Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research, 2018, pp. 569–576.
We present the results of the application of a parameter space reduction methodology based on active subspaces (AS) to the hull hydrodynamic design problem. Several parametric deformations of an initial hull shape are considered to assess the influence of the shape parameters on the hull wave resistance. Such problem is relevant at the preliminary stages of the ship design, when several flow simulations are carried out by the engineers to establish a certain sensibility with respect to the parameters, which might result in a high number of time consuming hydrodynamic simulations. The main idea of this work is to employ the AS to identify possible lower dimensional structures in the parameter space. The complete pipeline involves the use of free form deformation to parametrize and deform the hull shape, the high fidelity solver based on unsteady potential flow theory with fully nonlinear free surface treatment directly interfaced with CAD, the use of dynamic mode decomposition to reconstruct the final steady state given only few snapshots of the simulation, and the reduction of the parameter space by AS, and shared subspace. Response surface method is used to minimize the total drag.

@inproceedings{TezzeleDemoGadallaMolaRozza2018,
author = {Tezzele, Marco and Demo, Nicola and Gadalla, Mahmoud and Mola, Andrea and Rozza, Gianluigi},
booktitle = {Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship \& Maritime Research},
doi = {10.3233/978-1-61499-870-9-569},
pages = {569--576},
publisher = {IOS Press},
title = {Model order reduction by means of active subspaces and dynamic mode decomposition for parametric hull shape design hydrodynamics},
year = {2018},
preprint = {https://arxiv.org/abs/1803.07377},
abstract = {We present the results of the application of a parameter space reduction methodology based on active subspaces (AS) to the hull hydrodynamic design problem. Several parametric deformations of an initial hull shape are considered to assess the influence of the shape parameters on the hull wave resistance. Such problem is relevant at the preliminary stages of the ship design, when several flow simulations are carried out by the engineers to establish a certain sensibility with respect to the parameters, which might result in a high number of time consuming hydrodynamic simulations. The main idea of this work is to employ the AS to identify possible lower dimensional structures in the parameter space. The complete pipeline involves the use of free form deformation to parametrize and deform the hull shape, the high fidelity solver based on unsteady potential flow theory with fully nonlinear free surface treatment directly interfaced with CAD, the use of dynamic mode decomposition to reconstruct the final steady state given only few snapshots of the simulation, and the reduction of the parameter space by AS, and shared subspace. Response surface method is used to minimize the total drag.}
}

16. M. Tezzele, F. Salmoiraghi, A. Mola, and G. Rozza, "Dimension reduction in heterogeneous parametric spaces with application to naval engineering shape design problems", Advanced Modeling and Simulation in Engineering Sciences, 5(1), pp. 25, 2018.
We present the results of the first application in the naval architecture field of a methodology based on active subspaces properties for parameters space reduction. The physical problem considered is the one of the simulation of the hydrodynamic flow past the hull of a ship advancing in calm water. Such problem is extremely relevant at the preliminary stages of the ship design, when several flow simulations are typically carried out by the engineers to assess the dependence of the hull total resistance on the geometrical parameters of the hull, and others related with flows and hull properties. Given the high number of geometric and physical parameters which might affect the total ship drag, the main idea of this work is to employ the active subspaces properties to identify possible lower dimensional structures in the parameter space. Thus, a fully automated procedure has been implemented to produce several small shape perturbations of an original hull CAD geometry, in order to exploit the resulting shapes to run high fidelity flow simulations with different structural and physical parameters as well, and then collect data for the active subspaces analysis. The free form deformation procedure used to morph the hull shapes, the high fidelity solver based on potential flow theory with fully nonlinear free surface treatment, and the active subspaces analysis tool employed in this work have all been developed and integrated within SISSA mathLab as open source tools. The contribution will also discuss several details of the implementation of such tools, as well as the results of their application to the selected target engineering problem.

@article{TezzeleSalmoiraghiMolaRozza2018,
author = {Tezzele, Marco and Salmoiraghi, Filippo and Mola, Andrea and Rozza, Gianluigi},
title = {Dimension reduction in heterogeneous parametric spaces with application to naval engineering shape design problems},
journal = {Advanced Modeling and Simulation in Engineering Sciences},
number = {1},
pages = {25},
doi = {10.1186/s40323-018-0118-3},
preprint = {http://arxiv.org/abs/1709.03298},
volume = {5},
year = {2018},
abstract = {We present the results of the first application in the naval architecture field of a methodology based on active subspaces properties for parameters space reduction. The physical problem considered is the one of the simulation of the hydrodynamic flow past the hull of a ship advancing in calm water. Such problem is extremely relevant at the preliminary stages of the ship design, when several flow simulations are typically carried out by the engineers to assess the dependence of the hull total resistance on the geometrical parameters of the hull, and others related with flows and hull properties. Given the high number of geometric and physical parameters which might affect the total ship drag, the main idea of this work is to employ the active subspaces properties to identify possible lower dimensional structures in the parameter space. Thus, a fully automated procedure has been implemented to produce several small shape perturbations of an original hull CAD geometry, in order to exploit the resulting shapes to run high fidelity flow simulations with different structural and physical parameters as well, and then collect data for the active subspaces analysis. The free form deformation procedure used to morph the hull shapes, the high fidelity solver based on potential flow theory with fully nonlinear free surface treatment, and the active subspaces analysis tool employed in this work have all been developed and integrated within SISSA mathLab as open source tools. The contribution will also discuss several details of the implementation of such tools, as well as the results of their application to the selected target engineering problem.},
}

17. M. Tezzele, N. Demo, A. Mola, and G. Rozza, "An integrated data-driven computational pipeline with model order reduction for industrial and applied mathematics", 2018.
In this work we present an integrated computational pipeline involving several model order reduction techniques for industrial and applied mathematics, as emerging technology for product and/or process design procedures. Its data-driven nature and its modularity allow an easy integration into existing pipelines. We describe a complete optimization framework with automated geometrical parameterization, reduction of the dimension of the parameter space, and non-intrusive model order reduction such as dynamic mode decomposition and proper orthogonal decomposition with interpolation. Moreover several industrial examples are illustrated.

@unpublished{TezzeleDemoMolaRozza2018,
author = {Tezzele, Marco and Demo, Nicola and Mola, Andrea and Rozza, Gianluigi},
title = {An integrated data-driven computational pipeline with model order reduction for industrial and applied mathematics},
year = {2018},
preprint = {https://arxiv.org/abs/1810.12364},
abstract = {In this work we present an integrated computational pipeline involving several model order reduction techniques for industrial and applied mathematics, as emerging technology for product and/or process design procedures. Its data-driven nature and its modularity allow an easy integration into existing pipelines. We describe a complete optimization framework with automated geometrical parameterization, reduction of the dimension of the parameter space, and non-intrusive model order reduction such as dynamic mode decomposition and proper orthogonal decomposition with interpolation. Moreover several industrial examples are illustrated.}
}

18. M. Tezzele, F. Ballarin, and G. Rozza, "Combined parameter and model reduction of cardiovascular problems by means of active subspaces and POD-Galerkin methods", in Mathematical and Numerical Modeling of the Cardiovascular System and Applications, D. Boffi, L. F. Pavarino, G. Rozza, S. Scacchi, and C. Vergara (eds.), Cham: Springer International Publishing, pp. 185–207, 2018.
In this chapter we introduce a combined parameter and model reduction methodology and present its application to the efficient numerical estimation of a pressure drop in a set of deformed carotids. The aim is to simulate a wide range of possible occlusions after the bifurcation of the carotid. A parametric description of the admissible deformations, based on radial basis functions interpolation, is introduced. Since the parameter space may be very large, the first step in the combined reduction technique is to look for active subspaces in order to reduce the parameter space dimension. Then, we rely on model order reduction methods over the lower dimensional parameter subspace, based on a POD-Galerkin approach, to further reduce the required computational effort and enhance computational efficiency.

@inbook{TezzeleBallarinRozza2018,
author = {Tezzele, Marco and Ballarin, Francesco and Rozza, Gianluigi},
chapter = {Combined parameter and model reduction of cardiovascular problems by means of active subspaces and POD-Galerkin methods},
booktitle = {Mathematical and Numerical Modeling of the Cardiovascular System and Applications},
doi = {10.1007/978-3-319-96649-6_8},
editor = {Boffi, Daniele and Pavarino, Luca F. and Rozza, Gianluigi and Scacchi, Simone and Vergara, Christian},
isbn = {978-3-319-96649-6},
pages = {185--207},
publisher = {Springer International Publishing},
year = {2018},
preprint = {https://arxiv.org/abs/1711.10884},
abstract = {In this chapter we introduce a combined parameter and model reduction methodology and present its application to the efficient numerical estimation of a pressure drop in a set of deformed carotids. The aim is to simulate a wide range of possible occlusions after the bifurcation of the carotid. A parametric description of the admissible deformations, based on radial basis functions interpolation, is introduced. Since the parameter space may be very large, the first step in the combined reduction technique is to look for active subspaces in order to reduce the parameter space dimension. Then, we rely on model order reduction methods over the lower dimensional parameter subspace, based on a POD-Galerkin approach, to further reduce the required computational effort and enhance computational efficiency.}
}

19. D. Torlo, F. Ballarin, and G. Rozza, "Stabilized Weighted Reduced Basis Methods for Parametrized Advection Dominated Problems with Random Inputs", SIAM/ASA Journal on Uncertainty Quantification, 6(4), pp. 1475-1502, 2018.
In this work, we propose viable and efficient strategies for stabilized parametrized advection dominated problems, with random inputs. In particular, we investigate the combination of the wRB (weighted reduced basis) method for stochastic parametrized problems with the stabilized RB (reduced basis) method, which is the integration of classical stabilization methods (streamline/upwind Petrov–Galerkin (SUPG) in our case) in the offline–online structure of the RB method. Moreover, we introduce a reduction method that selectively enables online stabilization; this leads to a sensible reduction of computational costs, while keeping a very good accuracy with respect to high-fidelity solutions. We present numerical test cases to assess the performance of the proposed methods in steady and unsteady problems related to heat transfer phenomena.

@article{TorloBallarinRozza2018,
author = {Torlo, Davide and Ballarin, Francesco and Rozza, Gianluigi},
title = {Stabilized Weighted Reduced Basis Methods for Parametrized Advection Dominated Problems with Random Inputs},
journal = {SIAM/ASA Journal on Uncertainty Quantification},
volume = {6},
number = {4},
pages = {1475-1502},
year = {2018},
doi = {10.1137/17M1163517},
preprint = {https://arxiv.org/abs/1711.11275},
abstract = {In this work, we propose viable and efficient strategies for stabilized parametrized advection dominated problems, with random inputs. In particular, we investigate the combination of the wRB (weighted reduced basis) method for stochastic parametrized problems with the stabilized RB (reduced basis) method, which is the integration of classical stabilization methods (streamline/upwind Petrov--Galerkin (SUPG) in our case) in the offline--online structure of the RB method. Moreover, we introduce a reduction method that selectively enables online stabilization; this leads to a sensible reduction of computational costs, while keeping a very good accuracy with respect to high-fidelity solutions. We present numerical test cases to assess the performance of the proposed methods in steady and unsteady problems related to heat transfer phenomena.}
}

20. L. Venturi, F. Ballarin, and G. Rozza, "A Weighted POD Method for Elliptic PDEs with Random Inputs", Journal of Scientific Computing, 81(1), pp. 136–153, 2018.
In this work we propose and analyze a weighted proper orthogonal decomposition method to solve elliptic partial differential equations depending on random input data, for stochastic problems that can be transformed into parametric systems. The algorithm is introduced alongside the weighted greedy method. Our proposed method aims to minimize the error in a L2 norm and, in contrast to the weighted greedy approach, it does not require the availability of an error bound. Moreover, we consider sparse discretization of the input space in the construction of the reduced model; for high-dimensional problems, provided the sampling is done accordingly to the parameters distribution, this enables a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions. We provide many numerical tests to asses the performance of the proposed method compared to an equivalent reduced order model without weighting, as well as to the weighted greedy approach, in both low and higher dimensional problems.

@article{VenturiBallarinRozza2018,
author = {Venturi, Luca and Ballarin, Francesco and Rozza, Gianluigi},
title = {A Weighted POD Method for Elliptic PDEs with Random Inputs},
year = {2018},
preprint = {https://arxiv.org/abs/1802.08724},
journal={Journal of Scientific Computing},
volume={81},
number={1},
pages={136--153},
doi={10.1007/s10915-018-0830-7},
abstract = {In this work we propose and analyze a weighted proper orthogonal decomposition method to solve elliptic partial differential equations depending on random input data, for stochastic problems that can be transformed into parametric systems. The algorithm is introduced alongside the weighted greedy method. Our proposed method aims to minimize the error in a L2 norm and, in contrast to the weighted greedy approach, it does not require the availability of an error bound. Moreover, we consider sparse discretization of the input space in the construction of the reduced model; for high-dimensional problems, provided the sampling is done accordingly to the parameters distribution, this enables a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions.
We provide many numerical tests to asses the performance of the proposed method compared to an equivalent reduced order model without weighting, as well as to the weighted greedy approach, in both low and higher dimensional problems.}
}

### 2017

1. F. Ballarin, E. Faggiano, A. Manzoni, A. Quarteroni, G. Rozza, S. Ippolito, C. Antona, and R. Scrofani, "Numerical modeling of hemodynamics scenarios of patient-specific coronary artery bypass grafts", Biomechanics and Modeling in Mechanobiology, 16(4), pp. 1373–1399, 2017.
A fast computational framework is devised to the study of several configurations of patient-specific coronary artery bypass grafts. This is especially useful to perform a sensitivity analysis of the haemodynamics for different flow conditions occurring in native coronary arteries and bypass grafts, the investigation of the progression of the coronary artery disease and the choice of the most appropriate surgical procedure. A complete pipeline, from the acquisition of patientspecific medical images to fast parametrized computational simulations, is proposed. Complex surgical configurations employed in the clinical practice, such as Y-grafts and sequential grafts, are studied. A virtual surgery platform based on model reduction of unsteady Navier Stokes equations for blood dynamics is proposed to carry out sensitivity analyses in a very rapid and reliable way. A specialized geometrical parametrization is employed to compare the effect of stenosis and anastomosis variation on the outcome of the surgery in several relevant cases.

@ARTICLE{BallarinFaggianoManzoniQuarteroniRozzaIppolitoAntonaScrofani2016,
title = {Numerical modeling of hemodynamics scenarios of patient-specific coronary artery bypass grafts},
journal = {Biomechanics and Modeling in Mechanobiology},
abstract = {A fast computational framework is devised to the study of several configurations of patient-specific coronary artery bypass grafts. This is especially useful to perform a sensitivity analysis of the haemodynamics for different flow conditions occurring in native coronary arteries and bypass grafts, the investigation of the progression of the coronary artery disease and the choice of the most appropriate surgical procedure. A complete pipeline, from the acquisition of patientspecific medical images to fast parametrized computational simulations, is proposed. Complex surgical configurations employed in the clinical practice, such as Y-grafts and sequential grafts, are studied. A virtual surgery platform based on model reduction of unsteady Navier Stokes equations for blood dynamics is proposed to carry out sensitivity analyses in a very rapid and reliable way. A specialized geometrical parametrization is employed to compare the effect of stenosis and anastomosis variation on the outcome of the surgery in several relevant cases.},
author = {Francesco Ballarin and Elena Faggiano and Andrea Manzoni and Alfio Quarteroni and Gianluigi Rozza and Sonia Ippolito and Carlo Antona and Roberto Scrofani},
doi = {10.1007/s10237-017-0893-7},
year = {2017},
volume = {16},
number = {4},
pages = {1373--1399},
preprint = {https://preprints.sissa.it/xmlui/bitstream/handle/1963/35240/BMMB_SISSA_report.pdf?sequence=1&isAllowed=y}
}

2. F. Ballarin, G. Rozza, and Y. Maday, "Reduced-order semi-implicit schemes for fluid-structure interaction problems", in Model Reduction of Parametrized Systems, P. Benner, M. Ohlberger, A. Patera, G. Rozza, and K. Urban (eds.), Springer International Publishing, vol. 17, pp. 149–167, 2017.
POD–Galerkin reduced-order models (ROMs) for fluid-structure interaction problems (incompressible fluid and thin structure) are proposed in this paper. Both the high-fidelity and reduced-order methods are based on a Chorin-Temam operator-splitting approach. Two different reduced-order methods are proposed, which differ on velocity continuity condition, imposed weakly or strongly, respectively. The resulting ROMs are tested and compared on a representative haemodynamics test case characterized by wave propagation, in order to assess the capabilities of the proposed strategies.

@inbook{BallarinRozzaMaday2017,
chapter = {Reduced-order semi-implicit schemes for fluid-structure interaction problems},
year = {2017},
author = {Ballarin, Francesco and Rozza, Gianluigi and Maday, Yvon},
editor = {Benner, Peter and Ohlberger, Mario and Patera, Anthony and Rozza, Gianluigi and Urban, Karsten},
booktitle = {Model Reduction of Parametrized Systems},
publisher = {Springer International Publishing},
pages = {149--167},
volume = {17},
abstract = {POD--Galerkin reduced-order models (ROMs) for fluid-structure interaction problems (incompressible fluid and thin structure) are proposed in this paper. Both the high-fidelity and reduced-order methods are based on a Chorin-Temam operator-splitting approach. Two different reduced-order methods are proposed, which differ on velocity continuity condition, imposed weakly or strongly, respectively. The resulting ROMs are tested and compared on a representative haemodynamics test case characterized by wave propagation, in order to assess the capabilities of the proposed strategies.},
doi = {10.1007/978-3-319-58786-8_10},
preprint = {https://arxiv.org/abs/1711.10829}
}

3. T. Chacón Rebollo, E. Delgado Ávila, M. Gómez Mármol, F. Ballarin, and G. Rozza, "On a certified Smagorinsky reduced basis turbulence model", SIAM Journal on Numerical Analysis, 55(6), pp. 3047–3067, 2017.
In this work we present a reduced basis Smagorinsky turbulence model for steady flows. We approximate the non-linear eddy diffusion term using the Empirical Interpolation Method, and the velocity-pressure unknowns by an independent reduced-basis procedure. This model is based upon an a posteriori error estimation for Smagorinsky turbulence model. The theoretical development of the a posteriori error estimation is based on previous works, according to the Brezzi-Rappaz-Raviart stability theory, and adapted for the non-linear eddy diffusion term. We present some numerical tests, programmed in FreeFem++, in which we show an speedup on the computation by factor larger than 1000 in benchmark 2D flows.

@article{ChaconDelgadoGomezBallarinRozza2017,
author = {Chacón Rebollo, Tomás and Delgado Ávila, Enrique and Gómez Mármol, Macarena and Ballarin, Francesco and Rozza, Gianluigi},
title = {On a certified Smagorinsky reduced basis turbulence model},
year = {2017},
preprint = {https://arxiv.org/abs/1709.00243},
abstract = {In this work we present a reduced basis Smagorinsky turbulence model for steady flows. We approximate the non-linear eddy diffusion term using the Empirical Interpolation Method, and the velocity-pressure unknowns by an independent reduced-basis procedure. This model is based upon an a posteriori error estimation for Smagorinsky turbulence model. The theoretical development of the a posteriori error estimation is based on previous works, according to the Brezzi-Rappaz-Raviart stability theory, and adapted for the non-linear eddy diffusion term. We present some numerical tests, programmed in FreeFem++, in which we show an speedup on the computation by factor larger than 1000 in benchmark 2D flows.},
journal = {SIAM Journal on Numerical Analysis},
doi = {10.1137/17M1118233},
year = {2017},
volume = {55},
number = {6},
pages = {3047--3067},
}

4. P. Chen, A. Quarteroni, and G. Rozza, "Reduced Basis Methods for Uncertainty Quantification", SIAM/ASA Journal on Uncertainty Quantification, 5, pp. 813–869, 2017.
[BibTeX] [Abstract] [View on publisher website]
In this work we review a reduced basis method for the solution of uncertainty quantification problems. Based on the basic setting of an elliptic partial differential equation with random input, we introduce the key ingredients of the reduced basis method, including proper orthogonal decomposition and greedy algorithms for the construction of the reduced basis functions, a priori and a posteriori error estimates for the reduced basis approximations, as well as its computational advantages and weaknesses in comparison with a stochastic collocation method [I. Babuska, F. Nobile, and R. Tempone, SIAM Rev., 52 (2010), pp. 317–355]. We demonstrate its computational efficiency and accuracy for a benchmark problem with parameters ranging from a few to a few hundred dimensions. Generalizations to more complex models and applications to uncertainty quantification problems in risk prediction, evaluation of statistical moments, Bayesian inversion, and optimal control under uncertainty are also presented to illustrate how to use the reduced basis method in practice. Further challenges, advancements, and research opportunities are outlined.

@article{ChenQuarteroniRozza2017,
title = {Reduced Basis Methods for Uncertainty Quantification},
journal = {SIAM/ASA Journal on Uncertainty Quantification},
volume = {5},
year = {2017},
pages = {813--869},
doi = {10.1137/151004550},
abstract = {In this work we review a reduced basis method for the solution of uncertainty quantification problems. Based on the basic setting of an elliptic partial differential equation with random input, we introduce the key ingredients of the reduced basis method, including proper orthogonal decomposition and greedy algorithms for the construction of the reduced basis functions, a priori and a posteriori error estimates for the reduced basis approximations, as well as its computational advantages and weaknesses in comparison with a stochastic collocation method [I. Babuska, F. Nobile, and R. Tempone, SIAM Rev., 52 (2010), pp. 317--355]. We demonstrate its computational efficiency and accuracy for a benchmark problem with parameters ranging from a few to a few hundred dimensions. Generalizations to more complex models and applications to uncertainty quantification problems in risk prediction, evaluation of statistical moments, Bayesian inversion, and optimal control under uncertainty are also presented to illustrate how to use the reduced basis method in practice. Further challenges, advancements, and research opportunities are outlined.},
author = {Peng Chen and Alfio Quarteroni and Gianluigi Rozza}
}

5. F. Chinesta, A. Huerta, G. Rozza, and K. Willcox, "Model Reduction Methods", in Encyclopedia of Computational Mechanics Second Edition), John Wiley & Sons, pp. 1-36, 2017.
[BibTeX] [Abstract] [View on publisher website]
This chapter presents an overview of model order reduction – a new paradigm in the field of simulation-based engineering sciences, and one that can tackle the challenges and leverage the opportunities of modern ICT technologies. Despite the impressive progress attained by simulation capabilities and techniques, a number of challenging problems remain intractable. These problems are of different nature, but are common to many branches of science and engineering. Among them are those related to high-dimensional problems, problems involving very different time scales, models defined in degenerate domains with at least one of the characteristic dimensions much smaller than the others, model requiring real-time simulation, and parametric models. All these problems represent a challenge for standard mesh-based discretization techniques; yet the ability to solve these problems efficiently would open unexplored routes for real-time simulation, inverse analysis, uncertainty quantification and propagation, real-time optimization, and simulation-based control – critical needs in many branches of science and engineering. Model order reduction offers new simulation alternatives by circumventing, or at least alleviating, otherwise intractable computational challenges. In the present chapter, we revisit three of these model reduction techniques: proper orthogonal decomposition, proper generalized decomposition, and reduced basis methodologies.} preprint = {http://preprints.sissa.it/xmlui/bitstream/handle/1963/35194/ECM_MOR.pdf?sequence=1&isAllowed=y

@inbook{ChinestaHuertaRozzaWillcox2017,
author = {Chinesta, Francisco and Huerta, Antonio and Rozza, Gianluigi and Willcox, Karen},
publisher = {John Wiley \& Sons},
chapter = {Model Reduction Methods},
booktitle = {Encyclopedia of Computational Mechanics Second Edition},
pages = {1-36},
doi = {10.1002/9781119176817.ecm2110},
year = {2017},
abstract = {This chapter presents an overview of model order reduction – a new paradigm in the field of simulation-based engineering sciences, and one that can tackle the challenges and leverage the opportunities of modern ICT technologies. Despite the impressive progress attained by simulation capabilities and techniques, a number of challenging problems remain intractable. These problems are of different nature, but are common to many branches of science and engineering. Among them are those related to high-dimensional problems, problems involving very different time scales, models defined in degenerate domains with at least one of the characteristic dimensions much smaller than the others, model requiring real-time simulation, and parametric models. All these problems represent a challenge for standard mesh-based discretization techniques; yet the ability to solve these problems efficiently would open unexplored routes for real-time simulation, inverse analysis, uncertainty quantification and propagation, real-time optimization, and simulation-based control – critical needs in many branches of science and engineering. Model order reduction offers new simulation alternatives by circumventing, or at least alleviating, otherwise intractable computational challenges. In the present chapter, we revisit three of these model reduction techniques: proper orthogonal decomposition, proper generalized decomposition, and reduced basis methodologies.}
preprint = {http://preprints.sissa.it/xmlui/bitstream/handle/1963/35194/ECM_MOR.pdf?sequence=1&isAllowed=y}
}

6. D. Devaud and G. Rozza, "Certified Reduced Basis Method for Affinely Parametric Isogeometric Analysis NURBS Approximation", in Spectral and High Order Methods for Partial Differential Equations), Springer, vol. 119, 2017.
In this work we apply reduced basis methods for parametric PDEs to an isogeometric formulation based on NURBS. The motivation for this work is an integrated and complete work pipeline from CAD to parametrization of domain geometry, then from full order to certified reduced basis solution. IsoGeometric Analysis (IGA) is a growing research theme in scientific computing and computational mechanics, as well as reduced basis methods for parametric PDEs. Their combination enhances the solution of some class of problems, especially the ones characterized by parametrized geometries we introduced in this work. For a general overview on Reduced Basis (RB) methods we recall [7, 15] and on IGA [3]. This work wants to demonstrate that it is also possible for some class of problems to deal with ane geometrical parametrization combined with a NURBS IGA formulation. This is what this work brings as original ingredients with respect to other works dealing with reduced order methods and IGA (set in a non-affine formulation, and using a POD [2] sampling without certication: see for example for potential flows [12] and for Stokes flows [17]). In this work we show a certication of accuracy and a complete integration between IGA formulation and parametric certified greedy RB formulation.

@inbook{DevaudRozza2017,
chapter = {Certified Reduced Basis Method for Affinely Parametric Isogeometric Analysis NURBS Approximation},
booktitle = {Spectral and High Order Methods for Partial Differential Equations},
volume = {119},
year = {2017},
publisher = {Springer},
abstract = {In this work we apply reduced basis methods for parametric PDEs to an isogeometric formulation based on NURBS. The motivation for this work is an integrated and complete work pipeline from CAD to parametrization of domain geometry, then from full order to certified reduced basis solution. IsoGeometric Analysis (IGA) is a growing research theme in scientific computing and computational mechanics, as well as reduced basis methods for parametric PDEs. Their combination enhances the solution of some class of problems, especially the ones characterized by parametrized geometries we introduced in this work. For a general overview on Reduced Basis (RB) methods we recall [7, 15] and on IGA [3]. This work wants to demonstrate that it is also possible for some class of problems to deal with ane geometrical parametrization combined with a NURBS IGA formulation. This is what this work brings as original ingredients with respect to other works dealing with reduced order methods and IGA (set in a non-affine formulation, and using a POD [2] sampling without certication: see for example for potential flows [12] and for Stokes flows [17]). In this work we show a certication of accuracy and a complete integration between IGA formulation and parametric certified greedy RB formulation.},
author = {Devaud, Denis and Gianluigi Rozza},
preprint = {https://arxiv.org/abs/1710.06148},
doi = {10.1007/978-3-319-65870-4_3}
pages = {41--62},
}

7. S. Lorenzi, A. Cammi, L. Luzzi, and G. Rozza, "A reduced order model for investigating the dynamics of the Gen-IV LFR coolant pool", Applied Mathematical Modelling, 46, pp. 263-284, 2017.
[BibTeX] [View on publisher website]
@article{LorenziCammiLuzziRozza2017,
title = {A reduced order model for investigating the dynamics of the Gen-IV LFR coolant pool},
journal = {Applied Mathematical Modelling},
volume = {46},
year = {2017},
pages = {263-284},
doi = {10.1016/j.apm.2017.01.066},
author = {Lorenzi, S. and Cammi, A. and Luzzi, L. and Rozza, G.}
}

8. G. Pitton and G. Rozza, "On the Application of Reduced Basis Methods to Bifurcation Problems in Incompressible Fluid Dynamics", Journal of Scientific Computing, 73(1), pp. 157–177, 2017.
In this paper we apply a reduced basis framework for the computation of flow bifurcation (and stability) problems in fluid dynamics. The proposed method aims at reducing the complexity and the computational time required for the construction of bifurcation and stability diagrams. The method is quite general since it can in principle be specialized to a wide class of nonlinear problems, but in this work we focus on an application in incompressible fluid dynamics at low Reynolds numbers. The validation of the reduced order model with the full order computation for a benchmark cavity flow problem is promising.

@article{PittonRozza2017,
title = {On the Application of Reduced Basis Methods to Bifurcation Problems in Incompressible Fluid Dynamics},
journal = {Journal of Scientific Computing},
year = {2017},
abstract = {In this paper we apply a reduced basis framework for the computation of flow bifurcation (and stability) problems in fluid dynamics. The proposed method aims at reducing the complexity and the computational time required for the construction of bifurcation and stability diagrams. The method is quite general since it can in principle be specialized to a wide class of nonlinear problems, but in this work we focus on an application in incompressible fluid dynamics at low Reynolds numbers. The validation of the reduced order model with the full order computation for a benchmark cavity flow problem is promising.},
doi = {10.1007/s10915-017-0419-6},
volume = {73},
number = {1},
pages = {157--177},
author = {Giuseppe Pitton and Gianluigi Rozza},
preprint = {https://arxiv.org/abs/1801.00923}
}

9. G. Pitton, A. Quaini, and G. Rozza, "Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: Applications to Coanda effect in cardiology", Journal of Computational Physics, 344, pp. 534–557, 2017.
We focus on reducing the computational costs associated with the hydrodynamic stability of solutions of the incompressible Navier-Stokes equations for a Newtonian and viscous fluid in contraction-expansion channels. In particular, we are interested in studying steady bifurcations, occurring when non-unique stable solutions appear as physical and/or geometric control parameters are varied. The formulation of the stability problem requires solving an eigenvalue problem for a partial differential operator. An alternative to this approach is the direct simulation of the flow to characterize the asymptotic behavior of the solution. Both approaches can be extremely expensive in terms of computational time. We propose to apply Reduced Order Modeling (ROM) techniques to reduce the demanding computational costs associated with the detection of a type of steady bifurcations in fluid dynamics. The application that motivated the present study is the onset of asymmetries (i.e., symmetry breaking bifurcation) in blood flow through a regurgitant mitral valve, depending on the Reynolds number and the regurgitant mitral valve orifice shape.

@article{PittonQuainiRozza2017,
title = {Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: Applications to Coanda effect in cardiology},
journal = {Journal of Computational Physics},
volume = {344},
year = {2017},
pages = {534--557},
chapter = {534},
abstract = {We focus on reducing the computational costs associated with the hydrodynamic stability of solutions of the incompressible Navier-Stokes equations for a Newtonian and viscous fluid in contraction-expansion channels. In particular, we are interested in studying steady bifurcations, occurring when non-unique stable solutions appear as physical and/or geometric control parameters are varied. The formulation of the stability problem requires solving an eigenvalue problem for a partial differential operator. An alternative to this approach is the direct simulation of the flow to characterize the asymptotic behavior of the solution. Both approaches can be extremely expensive in terms of computational time. We propose to apply Reduced Order Modeling (ROM) techniques to reduce the demanding computational costs associated with the detection of a type of steady bifurcations in fluid dynamics. The application that motivated the present study is the onset of asymmetries (i.e., symmetry breaking bifurcation) in blood flow through a regurgitant mitral valve, depending on the Reynolds number and the regurgitant mitral valve orifice shape.},
doi = {10.1016/j.jcp.2017.05.010},
preprint = {https://arxiv.org/abs/1708.09718},
author = {Giuseppe Pitton and Annalisa Quaini and Gianluigi Rozza}
}

10. G. Stabile, S. N. Hijazi, S. Lorenzi, A. Mola, and G. Rozza, "POD-Galerkin Reduced Order Methods for CFD Using Finite Volume Discretisation: Vortex Shedding Around a Circular Cylinder", Communication in Applied Industrial Mathematics, 8(1), pp. 210–236, 2017.
Vortex shedding around circular cylinders is a well known and studied phenomenon that appears in many engineering fields. In this work a Reduced Order Model (ROM) of the incompressible flow around a circular cylinder, built performing a Galerkin projection of the governing equations onto a lower dimensional space is presented. The reduced basis space is generated using a Proper Orthogonal Decomposition (POD) approach. In particular the focus is into (i) the correct reproduction of the pressure field, that in case of the vortex shedding phenomenon, is of primary importance for the calculation of the drag and lift coefficients; (ii) for this purpose the projection of the Governing equations (momentum equation and Poisson equation for pressure) is performed onto different reduced basis space for velocity and pressure, respectively; (iii) all the relevant modifications necessary to adapt standard finite element POD-Galerkin methods to a finite volume framework are presented. The accuracy of the reduced order model is assessed against full order results.

@article{StabileHijaziLorenziMolaRozza2017,
title = {POD-Galerkin Reduced Order Methods for CFD Using Finite Volume Discretisation: Vortex Shedding Around a Circular Cylinder},
journal = {Communication in Applied Industrial Mathematics},
year = {2017},
volume = {8},
number = {1},
year = {2017},
pages = {210--236},
abstract = {Vortex shedding around circular cylinders is a well known and studied phenomenon that appears in many engineering fields. In this work a Reduced Order Model (ROM) of the incompressible flow around a circular cylinder, built performing a Galerkin projection of the governing equations onto a lower dimensional space is presented. The reduced basis space is generated using a Proper Orthogonal Decomposition (POD) approach. In particular the focus is into (i) the correct reproduction of the pressure field, that in case of the vortex shedding phenomenon, is of primary importance for the calculation of the drag and lift coefficients; (ii) for this purpose the projection of the Governing equations (momentum equation and Poisson equation for pressure) is performed onto different reduced basis space for velocity and pressure, respectively; (iii) all the relevant modifications necessary to adapt standard finite element POD-Galerkin methods to a finite volume framework are presented. The accuracy of the reduced order model is assessed against full order results.},
preprint = {https://arxiv.org/abs/1701.03424},
author = {Stabile, Giovanni and Hijazi, Saddam NY and Lorenzi, Stefano and Andrea Mola and Gianluigi Rozza},
doi = {10.1515/caim-2017-0011}
}

### 2016

1. F. Ballarin and G. Rozza, "POD–Galerkin monolithic reduced order models for parametrized fluid-structure interaction problems", International Journal for Numerical Methods in Fluids, 82(12), pp. 1010–1034, 2016.
In this paper we propose a monolithic approach for reduced order modelling of parametrized fluid-structure interaction problems based on a proper orthogonal decomposition (POD)–Galerkin method. Parameters of the problem are related to constitutive properties of the fluid or structural problem, or to geometrical parameters related to the domain configuration at the initial time. We provide a detailed description of the parametrized formulation of the multiphysics problem in its components, together with some insights on how to obtain an offline-online efficient computational procedure through the approximation of parametrized nonlinear tensors. Then, we present the monolithic POD–Galerkin method for the online computation of the global structural displacement, fluid velocity and pressure of the coupled problem. Finally, we show some numerical results to highlight the capabilities of the proposed reduced order method and its computational performances

@article{BallarinRozza2016,
author = {Francesco Ballarin and Gianluigi Rozza},
title = {{POD}--{G}alerkin monolithic reduced order models for parametrized fluid-structure interaction problems},
journal = {International Journal for Numerical Methods in Fluids},
volume = {82},
number = {12},
pages = {1010--1034},
abstract = {In this paper we propose a monolithic approach for reduced order modelling of parametrized fluid-structure interaction problems based on a proper orthogonal decomposition (POD)--Galerkin method. Parameters of the problem are related to constitutive properties of the fluid or structural problem, or to geometrical parameters related to the domain configuration at the initial time. We provide a detailed description of the parametrized formulation of the multiphysics problem in its components, together with some insights on how to obtain an offline-online efficient computational procedure through the approximation of parametrized nonlinear tensors. Then, we present the monolithic POD--Galerkin method for the online computation of the global structural displacement, fluid velocity and pressure of the coupled problem. Finally, we show some numerical results to highlight the capabilities of the proposed reduced order method and its computational performances},
year = {2016},
doi = {10.1002/fld.4252},
preprint = {http://preprints.sissa.it/xmlui/bitstream/handle/1963/35180/Navon75.pdf?sequence=1&isAllowed=y}
}

2. F. Ballarin, E. Faggiano, S. Ippolito, A. Manzoni, A. Quarteroni, G. Rozza, and R. Scrofani, "Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD-Galerkin method and a vascular shape parametrization", Journal of Computational Physics, 315, pp. 609–628, 2016.
In this work a reduced-order computational framework for the study of haemodynamics in three-dimensional patient-specific configurations of coronary artery bypass grafts dealing with a wide range of scenarios is proposed. We combine several efficient algorithms to face at the same time both the geometrical complexity involved in the description of the vascular network and the huge computational cost entailed by time dependent patient-specific flow simulations. Medical imaging procedures allow to reconstruct patient-specific configurations from clinical data. A centerlines-based parametrization is proposed to efficiently handle geometrical variations. POD–Galerkin reduced-order models are employed to cut down large computational costs. This computational framework allows to characterize blood flows for different physical and geometrical variations relevant in the clinical practice, such as stenosis factors and anastomosis variations, in a rapid and reliable way. Several numerical results are discussed, highlighting the computational performance of the proposed framework, as well as its capability to carry out sensitivity analysis studies, so far out of reach. In particular, a reduced-order simulation takes only a few minutes to run, resulting in computational savings of 99% of CPU time with respect to the full-order discretization. Moreover, the error between full-order and reduced-order solutions is also studied, and it is numerically found to be less than 1% for reduced-order solutions obtained with just O(100) online degrees of freedom.

@ARTICLE{BallarinFaggianoIppolitoManzoniQuarteroniRozzaScrofani2015,
author = {Ballarin, F. and Faggiano, E. and Ippolito, S. and Manzoni, A. and
Quarteroni, A. and Rozza, G. and Scrofani, R.},
title = {Fast simulations of patient-specific haemodynamics of coronary artery
bypass grafts based on a {POD}-{G}alerkin method and a vascular shape
parametrization},
year = {2016},
journal = {Journal of Computational Physics},
volume = {315},
pages = {609--628},
abstract = {In this work a reduced-order computational framework for the study of haemodynamics in three-dimensional patient-specific configurations of coronary artery bypass grafts dealing with a wide range of scenarios is proposed. We combine several efficient algorithms to face at the same time both the geometrical complexity involved in the description of the vascular network and the huge computational cost entailed by time dependent patient-specific flow simulations. Medical imaging procedures allow to reconstruct patient-specific configurations from clinical data. A centerlines-based parametrization is proposed to efficiently handle geometrical variations. POD--Galerkin reduced-order models are employed to cut down large computational costs. This computational framework allows to characterize blood flows for different physical and geometrical variations relevant in the clinical practice, such as stenosis factors and anastomosis variations, in a rapid and reliable way. Several numerical results are discussed, highlighting the computational performance of the proposed framework, as well as its capability to carry out sensitivity analysis studies, so far out of reach. In particular, a reduced-order simulation takes only a few minutes to run, resulting in computational savings of 99% of CPU time with respect to the full-order discretization. Moreover, the error between full-order and reduced-order solutions is also studied, and it is numerically found to be less than 1% for reduced-order solutions obtained with just O(100) online degrees of freedom.},
doi = {10.1016/j.jcp.2016.03.065},
preprint = {http://preprints.sissa.it/xmlui/bitstream/handle/1963/34623/REPORT.pdf?sequence=1&isAllowed=y}
}

3. L. Iapichino, A. Quarteroni, and G. Rozza, "Reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries", Computers and Mathematics with Applications, 71(1), pp. 408–430, 2016.
The aim of this work is to solve parametrized partial differential equations in computational domains represented by networks of repetitive geometries by combining reduced basis and domain decomposition techniques. The main idea behind this approach is to compute once, locally and for few reference shapes, some representative finite element solutions for different values of the parameters and with a set of different suitable boundary conditions on the boundaries: these functions will represent the basis of a reduced space where the global solution is sought for. The continuity of the latter is assured by a classical domain decomposition approach. Test results on Poisson problem show the flexibility of the proposed method in which accuracy and computational time may be tuned by varying the number of reduced basis functions employed, or the set of boundary conditions used for defining locally the basis functions. The proposed approach simplifies the pre-computation of the reduced basis space by splitting the global problem into smaller local subproblems. Thanks to this feature, it allows dealing with arbitrarily complex network and features more flexibility than a classical global reduced basis approximation where the topology of the geometry is fixed.

@ARTICLE{IapichinoQuarteroniRozza2016,
author = {Iapichino, L. and Quarteroni, A. and Rozza, G.},
title = {Reduced basis method and domain decomposition for elliptic problems
in networks and complex parametrized geometries},
journal = {Computers and Mathematics with Applications},
year = {2016},
volume = {71},
pages = {408--430},
number = {1},
abstract = {The aim of this work is to solve parametrized partial differential
equations in computational domains represented by networks of repetitive
geometries by combining reduced basis and domain decomposition techniques.
The main idea behind this approach is to compute once, locally and
for few reference shapes, some representative finite element solutions
for different values of the parameters and with a set of different
suitable boundary conditions on the boundaries: these functions will
represent the basis of a reduced space where the global solution
is sought for. The continuity of the latter is assured by a classical
domain decomposition approach. Test results on Poisson problem show
the flexibility of the proposed method in which accuracy and computational
time may be tuned by varying the number of reduced basis functions
employed, or the set of boundary conditions used for defining locally
the basis functions. The proposed approach simplifies the pre-computation
of the reduced basis space by splitting the global problem into smaller
local subproblems. Thanks to this feature, it allows dealing with
arbitrarily complex network and features more flexibility than a
classical global reduced basis approximation where the topology of
the geometry is fixed.},
doi = {10.1016/j.camwa.2015.12.001},
}

4. S. Lorenzi, A. Cammi, L. Luzzi, and G. Rozza, "POD-Galerkin Method for Finite Volume Approximation of Navier-Stokes and RANS Equations", Computer Methods in Applied Mechanics and Engineering, 311, pp. 151-179, 2016.
[BibTeX] [Abstract] [View on publisher website]
Numerical simulation of fluid flows requires important computational efforts but it is essential in engineering applications. Reduced Order Model (ROM) can be employed whenever fast simulations are required, or in general, whenever a trade-off between computational cost and solution accuracy is a preeminent issue as in process optimization and control. In this work, the efforts have been put to develop a ROM for Computational Fluid Dynamics (CFD) application based on Finite Volume approximation, starting from the results available in turbulent Reynold-Averaged Navier Stokes simulations in order to enlarge the application field of Proper Orthogonal Decomposition {–} Reduced Order Model (POD {–} ROM) technique to more industrial fields. The approach is tested in the classic benchmark of the numerical simulation of the 2D lid-driven cavity. In particular, two simulations at Re = 103 and Re = 105 have been considered in order to assess both a laminar and turbulent case. Some quantities have been compared with the Full Order Model in order to assess the performance of the proposed ROM procedure i.e., the kinetic energy of the system and the reconstructed quantities of interest (velocity, pressure and turbulent viscosity). In addition, for the laminar case, the comparison between the ROM steady-state solution and the data available in literature has been presented. The results have turned out to be very satisfactory both for the accuracy and the computational times. As a major outcome, the approach turns out not to be affected by the energy blow up issue characterizing the results obtained by classic turbulent POD-Galerkin methods.

@article{LorenziCammiLuzziRozza2016,
title = {{POD}-{G}alerkin Method for Finite Volume Approximation of Navier-Stokes and RANS Equations},
year = {2016},
volume = {311},
pages = {151 - 179},
year = {2016},
doi = {j.cma.2016.08.006},
journal = {Computer Methods in Applied Mechanics and Engineering},
abstract = {Numerical simulation of fluid flows requires important computational efforts but it is essential in engineering applications. Reduced Order Model (ROM) can be employed whenever fast simulations are required, or in general, whenever a trade-off between computational cost and solution accuracy is a preeminent issue as in process optimization and control. In this work, the efforts have been put to develop a ROM for Computational Fluid Dynamics (CFD) application based on Finite Volume approximation, starting from the results available in turbulent Reynold-Averaged Navier Stokes simulations in order to enlarge the application field of Proper Orthogonal Decomposition {\textendash} Reduced Order Model (POD {\textendash} ROM) technique to more industrial fields. The approach is tested in the classic benchmark of the numerical simulation of the 2D lid-driven cavity. In particular, two simulations at Re = 103 and Re = 105 have been considered in order to assess both a laminar and turbulent case. Some quantities have been compared with the Full Order Model in order to assess the performance of the proposed ROM procedure i.e., the kinetic energy of the system and the reconstructed quantities of interest (velocity, pressure and turbulent viscosity). In addition, for the laminar case, the comparison between the ROM steady-state solution and the data available in literature has been presented. The results have turned out to be very satisfactory both for the accuracy and the computational times. As a major outcome, the approach turns out not to be affected by the energy blow up issue characterizing the results obtained by classic turbulent POD-Galerkin methods.},
author = {Stefano Lorenzi and Antonio Cammi and Lelio Luzzi and Gianluigi Rozza}
}

5. F. Salmoiraghi, F. Ballarin, G. Corsi, A. Mola, M. Tezzele, and G. Rozza, "Advances in geometrical parametrization and reduced order models and methods for computational fluid dynamics problems in applied sciences and engineering: overview and perspectives", in Proceedings of the ECCOMAS Congress 2016, VII European Conference on Computational Methods in Applied Sciences and Engineering, 2016.
[BibTeX] [Abstract] [View on publisher website]
Several problems in applied sciences and engineering require reduction techniques in order to allow computational tools to be employed in the daily practice, especially in iterative procedures such as optimization or sensitivity analysis. Reduced order methods need to face increasingly complex problems in computational mechanics, especially into a multiphysics setting. Several issues should be faced: stability of the approximation, efficient treatment of nonlinearities, uniqueness or possible bifurcations of the state solutions, proper coupling between fields, as well as offline-online computing, computational savings and certification of errors as measure of accuracy. Moreover, efficient geometrical parametrization techniques should be devised to efficiently face shape optimization problems, as well as shape reconstruction and shape assimilation problems. A related aspect deals with the management of parametrized interfaces in multiphysics problems, such as fluid-structure interaction problems, and also a domain decomposition based approach for complex parametrized networks. We present some illustrative industrial and biomedical problems as examples of recent advances on methodological developments.

@inproceedings{SalmoiraghiBallarinCorsiMolaTezzeleRozza2016,
author = {Salmoiraghi, F. and Ballarin, F. and Corsi, G. and Mola, A. and Tezzele, M. and Rozza, G.},
title = {Advances in geometrical parametrization and reduced order models and methods for computational fluid dynamics problems in applied sciences and engineering: overview and perspectives},
booktitle = {Proceedings of the {ECCOMAS} {Congress} 2016, {VII} {E}uropean {C}onference on {C}omputational {M}ethods in {A}pplied {S}ciences and {E}ngineering},
year = {2016},
editor = {Papadrakakis, M. and Papadopoulos, V. and Stefanou, G. and Plevris, V.},
abstract = {Several problems in applied sciences and engineering require reduction techniques in order to allow computational tools to be employed in the daily practice, especially in iterative procedures such as optimization or sensitivity analysis. Reduced order methods need to face increasingly complex problems in computational mechanics, especially into a multiphysics setting. Several issues should be faced: stability of the approximation, efficient treatment of nonlinearities, uniqueness or possible bifurcations of the state solutions, proper coupling between fields, as well as offline-online computing, computational savings and certification of errors as measure of accuracy. Moreover, efficient geometrical parametrization techniques should be devised to efficiently face shape optimization problems, as well as shape reconstruction and shape assimilation problems. A related aspect deals with the management of parametrized interfaces in multiphysics problems, such as fluid-structure interaction problems, and also a domain decomposition based approach for complex parametrized networks. We present some illustrative industrial and biomedical problems as examples of recent advances on methodological developments.},
url = {http://www.eccomas.org/cvdata/cntr1/spc7/dtos/img/mdia/eccomas-2016-vol-1.pdf}
}

6. F. Salmoiraghi, F. Ballarin, L. Heltai, and G. Rozza, "Isogeometric analysis-based reduced order modelling for incompressible linear viscous flows in parametrized shapes", Advanced Modeling and Simulation in Engineering Sciences, 3(1), pp. 21, 2016.
[BibTeX] [Abstract] [View on publisher website]
In this work we provide a combination of isogeometric analysis with reduced order modelling techniques, based on proper orthogonal decomposition, to guarantee computational reduction for the numerical model, and with free-form deformation, for versatile geometrical parametrization. We apply it to computational fluid dynamics problems considering a Stokes flow model. The proposed reduced order model combines efficient shape deformation and accurate and stable velocity and pressure approximation for incompressible viscous flows, computed with a reduced order method. Efficient offine-online computational decomposition is guaranteed in view of repetitive calculations for parametric design and optimization problems. Numerical test cases show the efficiency and accuracy of the proposed reduced order model.

@article{SalmoiraghiBallarinHeltaiRozza2016,
title = {Isogeometric analysis-based reduced order modelling for incompressible linear viscous flows in parametrized shapes},
abstract = {In this work we provide a combination of isogeometric analysis with reduced order modelling techniques, based on proper orthogonal decomposition, to guarantee computational reduction for the numerical model, and with free-form deformation, for versatile geometrical parametrization. We apply it to computational fluid dynamics problems considering a Stokes flow model. The proposed reduced order model combines efficient shape deformation and accurate and stable velocity and pressure approximation
for incompressible viscous flows, computed with a reduced order method.
Efficient offine-online computational decomposition is guaranteed in view of repetitive calculations for parametric design and optimization problems.
Numerical test cases show the efficiency and accuracy of the proposed reduced order model.},
author = {Filippo Salmoiraghi and Francesco Ballarin and Luca Heltai and Gianluigi Rozza},
journal={Advanced Modeling and Simulation in Engineering Sciences},
year={2016},
volume={3},
number={1},
pages={21},
doi={10.1186/s40323-016-0076-6},
}

7. A. Sartori, A. Cammi, L. Luzzi, and G. Rozza, "A multi-physics reduced order model for the analysis of Lead Fast Reactor single channel", Annals of Nuclear Energy, 87, pp. 198–208, 2016.
[BibTeX] [Abstract] [View on publisher website]
In this work, a Reduced Basis method, with basis functions sampled by a Proper Orthogonal Decomposition technique, has been employed to develop a reduced order model of a multi-physics parametrized Lead-cooled Fast Reactor single-channel. Being the first time that a reduced order model is developed in this context, the work focused on a methodological approach and the coupling between the neutronics and the heat transfer, where the thermal feedbacks on neutronics are explicitly taken into account, in time-invariant settings. In order to address the potential of such approach, two different kinds of varying parameters have been considered, namely one related to a geometric quantity (i.e., the inner radius of the fuel pellet) and one related to a physical quantity (i.e., the inlet lead velocity). The capabilities of the presented reduced order model (ROM) have been tested and compared with a high-fidelity finite element model (upon which the ROM has been constructed) on different aspects. In particular, the comparison focused on the system reactivity prediction (with and without thermal feedbacks on neutronics), the neutron flux and temperature field reconstruction, and on the computational time. The outcomes provided by the reduced order model are in good agreement with the high-fidelity finite element ones, and a computational speed-up of at least three orders of magnitude is achieved as well.

@ARTICLE{SartoriCammiLuzziRozza2016,
author = {Sartori, A. and Cammi, A. and Luzzi, L. and Rozza, G.},
title = {A multi-physics reduced order model for the analysis of Lead Fast
Reactor single channel},
journal = {Annals of Nuclear Energy},
year = {2016},
volume = {87},
pages = {198--208},
abstract = {In this work, a Reduced Basis method, with basis functions sampled
by a Proper Orthogonal Decomposition technique, has been employed
to develop a reduced order model of a multi-physics parametrized
Lead-cooled Fast Reactor single-channel. Being the first time that
a reduced order model is developed in this context, the work focused
on a methodological approach and the coupling between the neutronics
and the heat transfer, where the thermal feedbacks on neutronics
are explicitly taken into account, in time-invariant settings. In
order to address the potential of such approach, two different kinds
of varying parameters have been considered, namely one related to
a geometric quantity (i.e., the inner radius of the fuel pellet)
and one related to a physical quantity (i.e., the inlet lead velocity).
The capabilities of the presented reduced order model (ROM) have
been tested and compared with a high-fidelity finite element model
(upon which the ROM has been constructed) on different aspects. In
particular, the comparison focused on the system reactivity prediction
(with and without thermal feedbacks on neutronics), the neutron flux
and temperature field reconstruction, and on the computational time.
The outcomes provided by the reduced order model are in good agreement
with the high-fidelity finite element ones, and a computational speed-up
of at least three orders of magnitude is achieved as well.},
doi = {10.1016/j.anucene.2015.09.002}
}

8. A. Sartori, A. Cammi, L. Luzzi, and G. Rozza, "A Reduced Basis Approach for Modeling the Movement of Nuclear Reactor Control Rods", ASME Journal of Nuclear Engineering and Radiation Science, 2(2), pp. 21019-8, 2016.
[BibTeX] [Abstract] [View on publisher website]
This work presents a reduced order model (ROM) aimed at simulating nuclear reactor control rods movement and featuring fast-running prediction of reactivity and neutron flux distribution as well. In particular, the reduced basis (RB) method (built upon a high-fidelity finite element (FE) approximation) has been employed. The neutronics has been modeled according to a parametrized stationary version of the multigroup neutron diffusion equation, which can be formulated as a generalized eigenvalue problem. Within the RB framework, the centroidal Voronoi tessellation is employed as a sampling technique due to the possibility of a hierarchical parameter space exploration, without relying on a “classical” a posteriori error estimation, and saving an important amount of computational time in the offline phase. Here, the proposed ROM is capable of correctly predicting, with respect to the high-fidelity FE approximation, both the reactivity and neutron flux shape. In this way, a computational speedup of at least three orders of magnitude is achieved. If a higher precision is required, the number of employed basis functions (BFs) must be increased

@ARTICLE{SartoriCammiLuzziRozza2016b,
author = {Sartori, A. and Cammi, A. and Luzzi, L. and Rozza, G.},
title = {A Reduced Basis Approach for Modeling the Movement of Nuclear Reactor Control Rods},
journal = {ASME Journal of Nuclear Engineering and Radiation Science},
year = {2016},
volume = {2},
number = {2},
pages = {021019-8},
abstract = {This work presents a reduced order model (ROM) aimed at simulating nuclear reactor control rods movement and featuring fast-running prediction of reactivity and neutron flux distribution as well. In particular, the reduced basis (RB) method (built upon a high-fidelity finite element (FE) approximation) has been employed. The neutronics has been modeled according to a parametrized stationary version of the multigroup neutron diffusion equation, which can be formulated as a generalized eigenvalue problem. Within the RB framework, the centroidal Voronoi tessellation is employed as a sampling technique due to the possibility of a hierarchical parameter space exploration, without relying on a “classical” a posteriori error estimation, and saving an important amount of computational time in the offline phase. Here, the proposed ROM is capable of correctly predicting, with respect to the high-fidelity FE approximation, both the reactivity and neutron flux shape. In this way, a computational speedup of at least three orders of magnitude is achieved. If a higher precision is required, the number of employed basis functions (BFs) must be increased},
doi = {10.1115/1.4031945}
}

9. A. Sartori, A. Cammi, L. Luzzi, and G. Rozza, "Reduced basis approaches in time-dependent non-coercive settings for modelling the movement of nuclear reactor control rods", Communications in Computational Physics, 20(1), pp. 23–59, 2016.
In this work, two approaches, based on the certified Reduced Basis method, have been developed for simulating the movement of nuclear reactor control rods, in time-dependent non-coercive settings featuring a 3D geometrical framework. In particular, in a first approach, a piece-wise affine transformation based on subdomains division has been implemented for modelling the movement of one control rod. In the second approach, a staircase strategy has been adopted for simulating the movement of all the three rods featured by the nuclear reactor chosen as case study. The neutron kinetics has been modelled according to the so-called multi-group neutron diffusion, which, in the present case, is a set of ten coupled parametrized parabolic equations (two energy groups for the neutron flux, and eight for the precursors). Both the reduced order models, developed according to the two approaches, provided a very good accuracy compared with high-fidelity results, assumed as truth solutions. At the same time, the computational speed-up in the Online phase, with respect to the fine truth finite element discretization, achievable by both the proposed approaches is at least of three orders of magnitude, allowing a real-time simulation of the rod movement and control.

@article{SartoriCammiLuzziRozza2015,
author = {Alberto Sartori and Antonio Cammi and Lelio Luzzi and Gianluigi Rozza},
title = {Reduced basis approaches in time-dependent non-coercive settings for
modelling the movement of nuclear reactor control rods},
abstract = {In this work, two approaches, based on the certified Reduced Basis
method, have been developed for simulating the movement of nuclear
reactor control rods, in time-dependent non-coercive settings featuring
a 3D geometrical framework. In particular, in a first approach, a
piece-wise affine transformation based on subdomains division has
been implemented for modelling the movement of one control rod. In
the second approach, a staircase strategy has been adopted for simulating
the movement of all the three rods featured by the nuclear reactor
chosen as case study. The neutron kinetics has been modelled according
to the so-called multi-group neutron diffusion, which, in the present
case, is a set of ten coupled parametrized parabolic equations (two
energy groups for the neutron flux, and eight for the precursors).
Both the reduced order models, developed according to the two approaches,
provided a very good accuracy compared with high-fidelity results,
assumed as truth solutions. At the same time, the computational speed-up
in the Online phase, with respect to the fine truth finite element
discretization, achievable by both the proposed approaches is at
least of three orders of magnitude, allowing a real-time simulation
of the rod movement and control.},
volume={20},
doi={10.4208/cicp.120914.021115a},
number={1},
journal={Communications in Computational Physics},
year={2016},
pages={23--59},
preprint = {https://iris.sissa.it/retrieve/handle/20.500.11767/15968/24759/manuscript.pdf}
}

### 2015

1. F. Ballarin, A. Manzoni, A. Quarteroni, and G. Rozza, "Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations", International Journal for Numerical Methods in Engineering, 102(5), pp. 1136–1161, 2015.
@ARTICLE{BallarinManzoniQuarteroniRozza2015,
author = {Ballarin, Francesco and Manzoni, Andrea and Quarteroni, Alfio and
Rozza, Gianluigi},
title = {Supremizer stabilization of {POD}--{G}alerkin approximation of parametrized
journal = {International Journal for Numerical Methods in Engineering},
year = {2015},
volume = {102},
pages = {1136--1161},
number = {5},
doi = {10.1002/nme.4772},
issn = {1097-0207},
}

2. P. Benner, M. Ohlberger, A. T. Patera, G. Rozza, D. C. Sorensen, and K. Urban, "Model order reduction of parameterized systems (MoRePaS): Preface to the special issue of advances in computational mathematics", Advances in Computational Mathematics, 41(5), pp. 955–960, 2015.
[BibTeX] [View on publisher website]
@ARTICLE{BennerOhlbergerPateraRozzaSorensenUrban2015,
author = {Benner, P. and Ohlberger, M. and Patera, A.T. and Rozza, G. and Sorensen,
D.C. and Urban, K.},
title = {Model order reduction of parameterized systems ({MoRePaS}): Preface
to the special issue of advances in computational mathematics},
journal = {Advances in Computational Mathematics},
year = {2015},
volume = {41},
pages = {955--960},
number = {5},
doi = {10.1007/s10444-015-9443-y}
}

3. P. Chen, A. Quarteroni, and G. Rozza, "Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations", Numerische Mathematik, 133(1), pp. 67–102, 2015.
In this paper we develop and analyze a multilevel weighted reduced basis method for solving stochastic optimal control problems constrained by Stokes equations. We prove the analytic regularity of the optimal solution in the probability space under certain assumptions on the random input data. The finite element method and the stochastic collocation method are employed for the numerical approximation of the problem in the deterministic space and the probability space, respectively, resulting in many large-scale optimality systems to solve. In order to reduce the unaffordable computational effort, we propose a reduced basis method using a multilevel greedy algorithm in combination with isotropic and anisotropic sparse-grid techniques. A weighted a posteriori error bound highlights the contribution stemming from each method. Numerical tests on stochastic dimensions ranging from 10 to 100 demonstrate that our method is very efficient, especially for solving high-dimensional and large-scale optimization problems.

@ARTICLE{ChenQuarteroniRozza2015,
author = {Chen, Peng and Quarteroni, Alfio and Rozza, Gianluigi},
title = {Multilevel and weighted reduced basis method for stochastic optimal
control problems constrained by {S}tokes equations},
journal = {Numerische Mathematik},
year = {2015},
volume = {133},
pages = {67--102},
number = {1},
abstract = {In this paper we develop and analyze a multilevel weighted reduced
basis method for solving stochastic optimal control problems constrained
by Stokes equations. We prove the analytic regularity of the optimal
solution in the probability space under certain assumptions on the
random input data. The finite element method and the stochastic collocation
method are employed for the numerical approximation of the problem
in the deterministic space and the probability space, respectively,
resulting in many large-scale optimality systems to solve. In order
to reduce the unaffordable computational effort, we propose a reduced
basis method using a multilevel greedy algorithm in combination with
isotropic and anisotropic sparse-grid techniques. A weighted a posteriori
error bound highlights the contribution stemming from each method.
Numerical tests on stochastic dimensions ranging from 10 to 100 demonstrate
that our method is very efficient, especially for solving high-dimensional
and large-scale optimization problems.},
doi = {10.1007/s00211-015-0743-4},
issn = {0945-3245},
preprint = {http://mathicse.epfl.ch/files/content/sites/mathicse/files/Mathicse%20reports%202013/33.2013_PC-AQ-GR.pdf}
}

4. D. Devaud and G. Rozza, "Reduced Basis Approximation for the Structural-Acoustic Design based on Energy Finite Element Analysis (RB-EFEA)", in CEMRACS 2013 - Modelling and simulation of complex systems: stochastic and deterministic approaches), , vol. 48, pp. 98–115, 2015.
[BibTeX] [View on publisher website]
@inbook{DevaudRozza2013,
chapter = {Reduced Basis Approximation for the Structural-Acoustic Design based on Energy Finite Element Analysis (RB-EFEA)},
booktitle = {CEMRACS 2013 - Modelling and simulation of complex systems: stochastic and deterministic approaches},
volume = {48},
number = {ESAIM Proceedings},
year = {2015},
pages = {98--115},
doi = {10.1051/proc/201448004},
author = {Denis Devaud and Gianluigi Rozza}
}

5. J. S. Hesthaven, G. Rozza, and B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations, 1 ed.), Switzerland: Springer, 2015.
[BibTeX] [Abstract] [View on publisher website]
This book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples.

@BOOK{HesthavenRozzaStamm2015,
title = {Certified Reduced Basis Methods for Parametrized Partial Differential
Equations},
publisher = {Springer},
year = {2015},
author = {Jan S. Hesthaven and Gianluigi Rozza and Benjamin Stamm},
pages = {135},
series = {Springer Briefs in Mathematics},
edition = {1},
abstract = {This book provides a thorough introduction to the mathematical and
algorithmic aspects of certified reduced basis methods for parametrized
partial differential equations. Central aspects ranging from model
construction, error estimation and computational efficiency to empirical
interpolation methods are discussed in detail for coercive problems.
More advanced aspects associated with time-dependent problems, non-compliant
and non-coercive problems and applications with geometric variation
are also discussed as examples.},
doi = {10.1007/978-3-319-22470-1},
isbn = {978-3-319-22469-5},
issn = {2191-8201},
organization = {Springer}
}

6. I. Martini, G. Rozza, and B. Haasdonk, "Reduced basis approximation and a-posteriori error estimation for the coupled Stokes-Darcy system", Advances in Computational Mathematics, 41(5), pp. 1131–1157, 2015.
The coupling of a free flow with a flow through porous media has many potential applications in several fields related with computational science and engineering, such as blood flows, environmental problems or food technologies. We present a reduced basis method for such coupled problems. The reduced basis method is a model order reduction method applied in the context of parametrized systems. Our approach is based on a heterogeneous domain decomposition formulation, namely the Stokes-Darcy problem. Thanks to an offline/online-decomposition, computational times can be drastically reduced. At the same time the induced error can be bounded by fast evaluable a-posteriori error bounds. In the offline-phase the proposed algorithms make use of the decomposed problem structure. Rigorous a-posteriori error bounds are developed, indicating the accuracy of certain lifting operators used in the offline-phase as well as the accuracy of the reduced coupled system. Also, a strategy separately bounding pressure and velocity errors is extended. Numerical experiments dealing with groundwater flow scenarios demonstrate the efficiency of the approach as well as the limitations regarding a-posteriori error estimation.

@ARTICLE{MartiniRozzaHaasdonk2015,
author = {Martini, Immanuel and Rozza, Gianluigi and Haasdonk, Bernard},
title = {Reduced basis approximation and a-posteriori error estimation for
the coupled {S}tokes-{D}arcy system},
journal = {Advances in Computational Mathematics},
year = {2015},
volume = {41},
pages = {1131--1157},
number = {5},
abstract = {The coupling of a free flow with a flow through porous media has many
potential applications in several fields related with computational
science and engineering, such as blood flows, environmental problems
or food technologies. We present a reduced basis method for such
coupled problems. The reduced basis method is a model order reduction
method applied in the context of parametrized systems. Our approach
is based on a heterogeneous domain decomposition formulation, namely
the Stokes-Darcy problem. Thanks to an offline/online-decomposition,
computational times can be drastically reduced. At the same time
the induced error can be bounded by fast evaluable a-posteriori error
bounds. In the offline-phase the proposed algorithms make use of
the decomposed problem structure. Rigorous a-posteriori error bounds
are developed, indicating the accuracy of certain lifting operators
used in the offline-phase as well as the accuracy of the reduced
coupled system. Also, a strategy separately bounding pressure and
velocity errors is extended. Numerical experiments dealing with groundwater
flow scenarios demonstrate the efficiency of the approach as well
as the limitations regarding a-posteriori error estimation.},
doi = {10.1007/s10444-014-9396-6},
issn = {1572-9044},
preprint = {http://preprints.sissa.it/xmlui/bitstream/handle/1963/34493/MRH14a_preprint.pdf?sequence=1&isAllowed=y}
}

7. F. Negri, A. Manzoni, and G. Rozza, "Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations", Computers and Mathematics with Applications, 69(4), pp. 319–336, 2015.
This paper extends the reduced basis method for the solution of parametrized optimal control problems presented in Negri et al. (2013) to the case of noncoercive (elliptic) equations, such as the Stokes equations. We discuss both the theoretical properties-with particular emphasis on the stability of the resulting double nested saddle-point problems and on aggregated error estimates-and the computational aspects of the method. Then, we apply it to solve a benchmark vorticity minimization problem for a parametrized bluff body immersed in a two or a three-dimensional flow through boundary control, demonstrating the effectivity of the methodology.

@ARTICLE{NegriManzoniRozza2015,
author = {Negri, F. and Manzoni, A. and Rozza, G.},
title = {Reduced basis approximation of parametrized optimal flow control
problems for the {S}tokes equations},
journal = {Computers and Mathematics with Applications},
year = {2015},
volume = {69},
pages = {319--336},
number = {4},
abstract = {This paper extends the reduced basis method for the solution of parametrized
optimal control problems presented in Negri et al. (2013) to the
case of noncoercive (elliptic) equations, such as the Stokes equations.
We discuss both the theoretical properties-with particular emphasis
on the stability of the resulting double nested saddle-point problems
and on aggregated error estimates-and the computational aspects of
the method. Then, we apply it to solve a benchmark vorticity minimization
problem for a parametrized bluff body immersed in a two or a three-dimensional
flow through boundary control, demonstrating the effectivity of the
methodology.},
doi = {10.1016/j.camwa.2014.12.010},
preprint = {http://mathicse.epfl.ch/files/content/sites/mathicse/files/Mathicse%20reports%202014/02.2014NEW-FNAMGR.pdf}
}

8. P. Pacciarini and G. Rozza, "Reduced basis approximation of parametrized advection-diffusion PDEs with high Péclet number", Lecture Notes in Computational Science and Engineering, 103, pp. 419–426, 2015.
In this work we show some results about the reduced basis approximation of advection dominated parametrized problems, i.e. advection-diffusion problems with high Péclet number. These problems are of great importance in several engineering applications and it is well known that their numerical approximation can be affected by instability phenomena. In this work we compare two possible stabilization strategies in the framework of the reduced basis method, by showing numerical results obtained for a steady advection-diffusion problem.

@ARTICLE{PacciariniRozza2015,
author = {Pacciarini, P. and Rozza, G.},
title = {Reduced basis approximation of parametrized advection-diffusion {PDE}s
with high {P}\'eclet number},
journal = {Lecture Notes in Computational Science and Engineering},
year = {2015},
volume = {103},
pages = {419--426},
abstract = {In this work we show some results about the reduced basis approximation
problems with high P\'eclet number. These problems are of great importance
in several engineering applications and it is well known that their
numerical approximation can be affected by instability phenomena.
In this work we compare two possible stabilization strategies in
the framework of the reduced basis method, by showing numerical results
doi = {10.1007/978-3-319-10705-9__41},
preprint = {https://infoscience.epfl.ch/record/203333/files/rozza_mini_ROMY.pdf}
}

### 2014

1. F. Ballarin, A. Manzoni, G. Rozza, and S. Salsa, "Shape Optimization by Free-Form Deformation: Existence Results and Numerical Solution for Stokes Flows", Journal of Scientific Computing, 60(3), pp. 537–563, 2014.
[BibTeX] [Abstract] [View on publisher website]
Shape optimization problems governed by PDEs result from many applications in computational fluid dynamics. These problems usually entail very large computational costs and require also a suitable approach for representing and deforming efficiently the shape of the underlying geometry, as well as for computing the shape gradient of the cost functional to be minimized. Several approaches based on the displacement of a set of control points have been developed in the last decades, such as the so-called free-form deformations. In this paper we present a new theoretical result which allows to recast free-form deformations into the general class of perturbation of identity maps, and to guarantee the compactness of the set of admissible shapes. Moreover, we address both a general optimization framework based on the continuous shape gradient and a numerical procedure for solving efficiently three-dimensional optimal design problems. This framework is applied to the optimal design of immersed bodies in Stokes flows, for which we consider the numerical solution of a benchmark case study from literature.

@ARTICLE{BallarinManzoniRozzaSalsa2014,
author = {Ballarin, F. and Manzoni, A. and Rozza, G. and Salsa, S.},
title = {Shape Optimization by Free-Form Deformation: Existence Results and
Numerical Solution for {S}tokes Flows},
journal = {Journal of Scientific Computing},
year = {2014},
volume = {60},
pages = {537--563},
number = {3},
abstract = {Shape optimization problems governed by PDEs result from many applications
in computational fluid dynamics. These problems usually entail very
large computational costs and require also a suitable approach for
representing and deforming efficiently the shape of the underlying
geometry, as well as for computing the shape gradient of the cost
functional to be minimized. Several approaches based on the displacement
of a set of control points have been developed in the last decades,
such as the so-called free-form deformations. In this paper we present
a new theoretical result which allows to recast free-form deformations
into the general class of perturbation of identity maps, and to guarantee
both a general optimization framework based on the continuous shape
gradient and a numerical procedure for solving efficiently three-dimensional
optimal design problems. This framework is applied to the optimal
design of immersed bodies in Stokes flows, for which we consider
the numerical solution of a benchmark case study from literature.},
doi = {10.1007/s10915-013-9807-8}
}

2. P. Chen, A. Quarteroni, and G. Rozza, "Comparison between reduced basis and stochastic collocation methods for elliptic problems", Journal of Scientific Computing, 59(1), pp. 187–216, 2014.
The stochastic collocation method (Babuška et al. in SIAM J Numer Anal 45(3):1005-1034, 2007; Nobile et al. in SIAM J Numer Anal 46(5):2411-2442, 2008a; SIAM J Numer Anal 46(5):2309-2345, 2008b; Xiu and Hesthaven in SIAM J Sci Comput 27(3):1118-1139, 2005) has recently been applied to stochastic problems that can be transformed into parametric systems. Meanwhile, the reduced basis method (Maday et al. in Comptes Rendus Mathematique 335(3):289-294, 2002; Patera and Rozza in Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations Version 1.0. Copyright MIT, http://augustine.mit.edu, 2007; Rozza et al. in Arch Comput Methods Eng 15(3):229-275, 2008), primarily developed for solving parametric systems, has been recently used to deal with stochastic problems (Boyaval et al. in Comput Methods Appl Mech Eng 198(41-44):3187-3206, 2009; Arch Comput Methods Eng 17:435-454, 2010). In this work, we aim at comparing the performance of the two methods when applied to the solution of linear stochastic elliptic problems. Two important comparison criteria are considered: (1), convergence results of the approximation error; (2), computational costs for both offline construction and online evaluation. Numerical experiments are performed for problems from low dimensions O(1) to moderate dimensions O(10) and to high dimensions O(100). The main result stemming from our comparison is that the reduced basis method converges better in theory and faster in practice than the stochastic collocation method for smooth problems, and is more suitable for large scale and high dimensional stochastic problems when considering computational costs.

@ARTICLE{ChenQuarteroniRozza2014,
author = {Chen, P. and Quarteroni, A. and Rozza, G.},
title = {Comparison between reduced basis and stochastic collocation methods
for elliptic problems},
journal = {Journal of Scientific Computing},
year = {2014},
volume = {59},
pages = {187--216},
number = {1},
abstract = {The stochastic collocation method (Babu\v{s}ka et al. in SIAM J Numer
Anal 45(3):1005-1034, 2007; Nobile et al. in SIAM J Numer Anal 46(5):2411-2442,
2008a; SIAM J Numer Anal 46(5):2309-2345, 2008b; Xiu and Hesthaven
in SIAM J Sci Comput 27(3):1118-1139, 2005) has recently been applied
to stochastic problems that can be transformed into parametric systems.
Meanwhile, the reduced basis method (Maday et al. in Comptes Rendus
Mathematique 335(3):289-294, 2002; Patera and Rozza in Reduced basis
approximation and a posteriori error estimation for parametrized
partial differential equations Version 1.0. Copyright MIT, http://augustine.mit.edu,
2007; Rozza et al. in Arch Comput Methods Eng 15(3):229-275, 2008),
primarily developed for solving parametric systems, has been recently
used to deal with stochastic problems (Boyaval et al. in Comput Methods
Appl Mech Eng 198(41-44):3187-3206, 2009; Arch Comput Methods Eng
17:435-454, 2010). In this work, we aim at comparing the performance
of the two methods when applied to the solution of linear stochastic
elliptic problems. Two important comparison criteria are considered:
(1), convergence results of the approximation error; (2), computational
costs for both offline construction and online evaluation. Numerical
experiments are performed for problems from low dimensions O(1) to
moderate dimensions O(10) and to high dimensions O(100). The main
result stemming from our comparison is that the reduced basis method
converges better in theory and faster in practice than the stochastic
collocation method for smooth problems, and is more suitable for
large scale and high dimensional stochastic problems when considering
computational costs.},
doi = {10.1007/s10915-013-9764-2},
preprint = {http://mathicse.epfl.ch/files/content/sites/mathicse/files/Mathicse%20reports%202012/34.2012_PC-AQ-GR.pdf}
}

3. P. Chen, A. Quarteroni, and G. Rozza, "A weighted empirical interpolation method: A priori convergence analysis and applications", ESAIM: Mathematical Modelling and Numerical Analysis, 48(4), pp. 943–953, 2014.
We extend the classical empirical interpolation method [M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Compt. Rend. Math. Anal. Num. 339 (2004) 667-672] to a weighted empirical interpolation method in order to approximate nonlinear parametric functions with weighted parameters, e.g. random variables obeying various probability distributions. A priori convergence analysis is provided for the proposed method and the error bound by Kolmogorov N-width is improved from the recent work [Y. Maday, N.C. Nguyen, A.T. Patera and G.S.H. Pau, A general, multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2009) 383-404]. We apply our method to geometric Brownian motion, exponential Karhunen-Loève expansion and reduced basis approximation of non-affine stochastic elliptic equations. We demonstrate its improved accuracy and efficiency over the empirical interpolation method, as well as sparse grid stochastic collocation method.

@ARTICLE{ChenQuarteroniRozza2014a,
author = {Chen, P. and Quarteroni, A. and Rozza, G.},
title = {A weighted empirical interpolation method: A priori convergence analysis
and applications},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
year = {2014},
volume = {48},
pages = {943--953},
number = {4},
abstract = {We extend the classical empirical interpolation method [M. Barrault,
Y. Maday, N.C. Nguyen and A.T. Patera, An empirical interpolation
method: application to efficient reduced-basis discretization of
partial differential equations. Compt. Rend. Math. Anal. Num. 339
(2004) 667-672] to a weighted empirical interpolation method in order
to approximate nonlinear parametric functions with weighted parameters,
e.g. random variables obeying various probability distributions.
A priori convergence analysis is provided for the proposed method
and the error bound by Kolmogorov N-width is improved from the recent
work [Y. Maday, N.C. Nguyen, A.T. Patera and G.S.H. Pau, A general,
multipurpose interpolation procedure: the magic points. Commun. Pure
Appl. Anal. 8 (2009) 383-404]. We apply our method to geometric Brownian
motion, exponential Karhunen-Lo\eve expansion and reduced basis
approximation of non-affine stochastic elliptic equations. We demonstrate
its improved accuracy and efficiency over the empirical interpolation
method, as well as sparse grid stochastic collocation method.},
doi = {10.1051/m2an/2013128},
preprint = {https://infoscience.epfl.ch/record/197090/files/05.2013_NEW_PC-AQ-GR.pdf}
}

4. D. Forti and G. Rozza, "Efficient geometrical parametrisation techniques of interfaces for reduced-order modelling: application to fluid–structure interaction coupling problems", International Journal of Computational Fluid Dynamics, 28(3-4), pp. 158–169, 2014.
[BibTeX] [Abstract] [View on publisher website]
We present some recent advances and improvements in shape parametrisation techniques of interfaces for reduced-order modelling with special attention to fluid–structure interaction problems and the management of structural deformations, namely, to represent them into a low-dimensional space (by control points). This allows to reduce the computational effort, and to significantly simplify the (geometrical) deformation procedure, leading to more efficient and fast reduced-order modelling applications in this kind of problems. We propose an efficient methodology to select the geometrical control points for the radial basis functions based on a modal greedy algorithm to improve the computational efficiency in view of more complex fluid–structure applications in several fields. The examples provided deal with aeronautics and wind engineering.

@ARTICLE{FortiRozza2014,
author = {Forti, D. and Rozza, G.},
title = {Efficient geometrical parametrisation techniques of interfaces for
reduced-order modelling: application to fluid--structure interaction
coupling problems},
journal = {International Journal of Computational Fluid Dynamics},
year = {2014},
volume = {28},
pages = {158--169},
number = {3-4},
abstract = {We present some recent advances and improvements in shape parametrisation
techniques of interfaces for reduced-order modelling with special
attention to fluid--structure interaction problems and the management
of structural deformations, namely, to represent them into a low-dimensional
space (by control points). This allows to reduce the computational
effort, and to significantly simplify the (geometrical) deformation
procedure, leading to more efficient and fast reduced-order modelling
applications in this kind of problems. We propose an efficient methodology
to select the geometrical control points for the radial basis functions
based on a modal greedy algorithm to improve the computational efficiency
in view of more complex fluid--structure applications in several
fields. The examples provided deal with aeronautics and wind engineering.},
doi = {10.1080/10618562.2014.932352}
}

5. L. Iapichino, A. Quarteroni, G. Rozza, and S. Volkwein, "Reduced basis method for the Stokes equations in decomposable domains using greedy optimization", in ECMI 2014 proceedings, 2014, pp. 1–7.
@INPROCEEDINGS{IapichinoQuarteroniRozzaVolkwein2014,
author = {Iapichino, Laura and Quarteroni, Alfio and Rozza, Gianluigi and Volkwein,
Stefan},
title = {Reduced basis method for the {S}tokes equations in decomposable domains
using greedy optimization},
year = {2014},
pages = {1--7},
booktitle = {ECMI 2014 proceedings},
preprint = {http://kops.uni-konstanz.de/handle/123456789/27996}
}

6. C. Jäggli, L. Iapichino, and G. Rozza, "An improvement on geometrical parameterizations by transfinite maps", Comptes Rendus Mathematique, 352(3), pp. 263–268, 2014.
We present a method to generate a non-affine transfinite map from a given reference domain to a family of deformed domains. The map is a generalization of the Gordon-Hall transfinite interpolation approach. It is defined globally over the reference domain. Once we have computed some functions over the reference domain, the map can be generated by knowing the parametric expressions of the boundaries of the deformed domain. Being able to define a suitable map from a reference domain to a desired deformation is useful for the management of parameterized geometries.

@ARTICLE{JaggliIapichinoRozza2014,
author = {J\"aggli, C. and Iapichino, L. and Rozza, G.},
title = {An improvement on geometrical parameterizations by transfinite maps},
journal = {Comptes Rendus Mathematique},
year = {2014},
volume = {352},
pages = {263--268},
number = {3},
abstract = {We present a method to generate a non-affine transfinite map from
a given reference domain to a family of deformed domains. The map
is a generalization of the Gordon-Hall transfinite interpolation
approach. It is defined globally over the reference domain. Once
we have computed some functions over the reference domain, the map
can be generated by knowing the parametric expressions of the boundaries
of the deformed domain. Being able to define a suitable map from
a reference domain to a desired deformation is useful for the management
of parameterized geometries.},
doi = {10.1016/j.crma.2013.12.017},
preprint = {http://preprints.sissa.it/xmlui/bitstream/handle/1963/35068/42.2013_CJ-LI-GR.pdf?sequence=2&isAllowed=y}
}

7. T. Lassila, A. Manzoni, A. Quarteroni, and G. Rozza, "Model order reduction in fluid dynamics: challenges and perspectives", in Reduced Order Methods for Modeling and Computational Reduction, A. Quarteroni and G. Rozza (eds.), Springer MS&A Series, vol. 9, pp. 235–274, 2014.
This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities - which are mainly related either to nonlinear convection terms and/or some geometric variability - that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and-in the unsteady case - long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.

@INCOLLECTION{LassilaManzoniQuarteroniRozza2014,
author = {Lassila, T. and Manzoni, A. and Quarteroni, A. and Rozza, G.},
title = {Model order reduction in fluid dynamics: challenges and perspectives},
booktitle = {Reduced Order Methods for Modeling and Computational Reduction},
publisher = {Springer MS\&A Series},
year = {2014},
editor = {A. Quarteroni and G. Rozza},
volume = {9},
pages = {235--274},
abstract = {This chapter reviews techniques of model reduction of fluid dynamics
systems. Fluid systems are known to be difficult to reduce efficiently
due to several reasons. First of all, they exhibit strong nonlinearities
- which are mainly related either to nonlinear convection terms and/or
some geometric variability - that often cannot be treated by simple
linearization. Additional difficulties arise when attempting model
reduction of unsteady flows, especially when long-term transient
behavior needs to be accurately predicted using reduced order models
and more complex features, such as turbulence or multiphysics phenomena,
have to be taken into consideration. We first discuss some general
principles that apply to many parametric model order reduction problems,
by the incompressible Navier-Stokes equations. We address questions
of inf-sup stability, certification through error estimation, computational
issues and-in the unsteady case - long-time stability of the reduced
model. Moreover, we provide an extensive list of literature references.},
doi = {10.1007/978-3-319-02090-7_9},
preprint = {https://infoscience.epfl.ch/record/187600/files/LMQR_ROMReview.pdf}
}

8. A. Manzoni, T. Lassila, A. Quarteroni, and G. Rozza, "A Reduced-Order Strategy for Solving Inverse Bayesian Shape Identification Problems in Physiological Flows", , G. H. Bock, P. X. Hoang, R. Rannacher, and J. P. Schlöder (eds.), Springer International Publishing, pp. 145–155, 2014.
@INBOOK{ManzoniLassilaQuarteroniRozza2014,
chapter = {A Reduced-Order Strategy for Solving Inverse Bayesian Shape Identification
Problems in Physiological Flows},
pages = {145--155},
title = {Modeling, Simulation and Optimization of Complex Processes - HPSC
2012: Proceedings of the Fifth International Conference on High Performance
Scientific Computing, March 5-9, 2012, Hanoi, Vietnam},
publisher = {Springer International Publishing},
year = {2014},
editor = {Bock, Georg Hans and Hoang, Phu Xuan and Rannacher, Rolf and Schl{\"o}der,
P. Johannes},
author = {Manzoni, Andrea and Lassila, Toni and Quarteroni, Alfio and Rozza,
Gianluigi},
doi = {10.1007/978-3-319-09063-4_12},
isbn = {978-3-319-09063-4},
preprint = {http://mathicse.epfl.ch/files/content/sites/mathicse/files/Mathicse%20reports%202012/19.2012_AM-TL-AQ-GR.pdf}
}

9. P. Pacciarini and G. Rozza, "Stabilized reduced basis method for parametrized advection-diffusion PDEs", Computer Methods in Applied Mechanics and Engineering, 274, pp. 1–18, 2014.
In this work, we propose viable and efficient strategies for the stabilization of the reduced basis approximation of an advection dominated problem. In particular, we investigate the combination of a classic stabilization method (SUPG) with the Offline-Online structure of the RB method. We explain why the stabilization is needed in both stages and we identify, analytically and numerically, which are the drawbacks of a stabilization performed only during the construction of the reduced basis (i.e. only in the Offline stage). We carry out numerical tests to assess the performances of the double'' stabilization both in steady and unsteady problems, also related to heat transfer phenomena.

@ARTICLE{PacciariniRozza2014,
author = {Pacciarini, P. and Rozza, G.},
title = {Stabilized reduced basis method for parametrized advection-diffusion
{PDE}s},
journal = {Computer Methods in Applied Mechanics and Engineering},
year = {2014},
volume = {274},
pages = {1--18},
abstract = {In this work, we propose viable and efficient strategies for the stabilization
of the reduced basis approximation of an advection dominated problem.
In particular, we investigate the combination of a classic stabilization
method (SUPG) with the Offline-Online structure of the RB method.
We explain why the stabilization is needed in both stages and we
identify, analytically and numerically, which are the drawbacks of
a stabilization performed only during the construction of the reduced
basis (i.e. only in the Offline stage). We carry out numerical tests
to assess the performances of the double'' stabilization both in
doi = {10.1016/j.cma.2014.02.005},
preprint = {http://mathicse.epfl.ch/files/content/sites/mathicse/files/Mathicse%20reports%202013/43.2013_PP-GR.pdf}
}

10. P. Pacciarini and G. Rozza, "Stabilized reduced basis method for parametrized scalar advection-diffusion problems at higher Péclet number: Roles of the boundary layers and inner fronts", in 11th World Congress on Computational Mechanics, WCCM 2014, 5th European Conference on Computational Mechanics, ECCM 2014 and 6th European Conference on Computational Fluid Dynamics, ECFD 2014, 2014, pp. 5614–5624.
Advection-dominated problems, which arise in many engineering situations, often require a fast and reliable approximation of the solution given some parameters as inputs. In this work we want to investigate the coupling of the reduced basis method - which guarantees rapidity and reliability - with some classical stabilization techiques to deal with the advection-dominated condition. We provide a numerical extension of the results presented in [1], focusing in particular on problems with curved boundary layers and inner fronts whose direction depends on the parameter.

@INPROCEEDINGS{PacciariniRozza2014a,
author = {Pacciarini, P. and Rozza, G.},
title = {Stabilized reduced basis method for parametrized scalar advection-diffusion
problems at higher {P}\'eclet number: Roles of the boundary layers
and inner fronts},
year = {2014},
pages = {5614--5624},
abstract = {Advection-dominated problems, which arise in many engineering situations,
often require a fast and reliable approximation of the solution given
some parameters as inputs. In this work we want to investigate the
coupling of the reduced basis method - which guarantees rapidity
and reliability - with some classical stabilization techiques to
deal with the advection-dominated condition. We provide a numerical
extension of the results presented in [1], focusing in particular
on problems with curved boundary layers and inner fronts whose direction
depends on the parameter.},
booktitle = {11th World Congress on Computational Mechanics, WCCM 2014, 5th European
Conference on Computational Mechanics, ECCM 2014 and 6th European
Conference on Computational Fluid Dynamics, ECFD 2014},
preprint = {https://infoscience.epfl.ch/record/203327/files/ECCOMAS_PP_GR.pdf},
url = {https://infoscience.epfl.ch/record/203327/files/ECCOMAS_PP_GR.pdf}
}

11. A. Quarteroni and G. Rozza, Reduced Order Methods for Modeling and Computational Reduction, 1 ed.), Springer, 2014, vol. 9.
[BibTeX] [Abstract] [View on publisher website]
This monograph addresses the state of the art of reduced order methods for modeling and computational reduction of complex parametrized systems, governed by ordinary and/or partial differential equations, with a special emphasis on real time computing techniques and applications in computational mechanics, bioengineering and computer graphics.Several topics are covered, including: design, optimization, and control theory in real-time with applications in engineering; data assimilation, geometry registration, and parameter estimation with special attention to real-time computing in biomedical engineering and computational physics; real-time visualization of physics-based simulations in computer science; the treatment of high-dimensional problems in state space, physical space, or parameter space; the interactions between different model reduction and dimensionality reduction approaches; the development of general error estimation frameworks which take into account both model and discretization effects. This book is primarily addressed to computational scientists interested in computational reduction techniques for large scale differential problems.

@BOOK{QuarteroniRozza2014,
title = {Reduced Order Methods for Modeling and Computational Reduction},
publisher = {Springer},
year = {2014},
author = {Alfio Quarteroni and Gianluigi Rozza},
volume = {9},
pages = {334},
series = {MS\&A},
edition = {1},
abstract = {This monograph addresses the state of the art of reduced order methods
for modeling and computational reduction of complex parametrized
systems, governed by ordinary and/or partial differential equations,
with a special emphasis on real time computing techniques and applications
in computational mechanics, bioengineering and computer graphics.Several
topics are covered, including: design, optimization, and control
theory in real-time with applications in engineering; data assimilation,
geometry registration, and parameter estimation with special attention
to real-time computing in biomedical engineering and computational
physics; real-time visualization of physics-based simulations in
computer science; the treatment of high-dimensional problems in state
space, physical space, or parameter space; the interactions between
different model reduction and dimensionality reduction approaches;
the development of general error estimation frameworks which take
into account both model and discretization effects. This book is
primarily addressed to computational scientists interested in computational
reduction techniques for large scale differential problems.},
doi = {10.1007/978-3-319-02090-7},
issn = {978-3-319-02089-1},
organization = {Springer}
}

12. G. Rozza, "Fundamentals of Reduced Basis Method for problems governed by parametrized PDEs and applications", in Separated representations and PGD-based model reduction: fundamentals and applications), Springer, vol. 554, 2014.
[BibTeX] [Abstract] [View on publisher website]
In this chapter we consider Reduced Basis (RB) approximations of parametrized Partial Differential Equations (PDEs). The the idea behind RB is to decouple the generation and projection stages (Offline/Online computational procedures) of the approximation process in order to solve parametrized PDEs in a fast, inexpensive and reliable way. The RB method, especially applied to 3D problems, allows great computational savings with respect to the classical Galerkin Finite Element (FE) Method. The standard FE method is typically ill suited to (i) iterative contexts like optimization, sensitivity analysis and many-queries in general, and (ii) real time evaluation. We consider for simplicity coercive PDEs. We discuss all the steps to set up a RB approximation, either from an analytical and a numerical point of view. Then we present an application of the RB method to a steady thermal conductivity problem in heat transfer with emphasis on geometrical and physical parameters.

@INBOOK{Rozza2014,
chapter = {Fundamentals of Reduced Basis Method for problems governed by parametrized
PDEs and applications},
publisher = {Springer},
year = {2014},
author = {Gianluigi Rozza},
volume = {554},
series = {CISM International Centre for Mechanical Sciences},
abstract = {In this chapter we consider Reduced Basis (RB) approximations of parametrized
Partial Differential Equations (PDEs). The the idea behind RB is
to decouple the generation and projection stages (Offline/Online
computational procedures) of the approximation process in order to
solve parametrized PDEs in a fast, inexpensive and reliable way.
The RB method, especially applied to 3D problems, allows great computational
savings with respect to the classical Galerkin Finite Element (FE)
Method. The standard FE method is typically ill suited to (i) iterative
contexts like optimization, sensitivity analysis and many-queries
in general, and (ii) real time evaluation. We consider for simplicity
coercive PDEs. We discuss all the steps to set up a RB approximation,
either from an analytical and a numerical point of view. Then we
present an application of the RB method to a steady thermal conductivity
problem in heat transfer with emphasis on geometrical and physical
parameters.},
booktitle = {Separated representations and {PGD}-based model reduction: fundamentals
and applications},
doi = {10.1007/978-3-7091-1794-1_4},
organization = {Springer}
}

13. A. Sartori, D. Baroli, A. Cammi, D. Chiesa, L. Luzzi, R. Ponciroli, E. Previtali, M. E. Ricotti, G. Rozza, and M. Sisti, "Comparison of a Modal Method and a Proper Orthogonal Decomposition approach for multi-group time-dependent reactor spatial kinetics", Annals of Nuclear Energy, 71, pp. 217–229, 2014.
[BibTeX] [Abstract] [View on publisher website]
In this paper, two modelling approaches based on a Modal Method (MM) and on the Proper Orthogonal Decomposition (POD) technique, for developing a control-oriented model of nuclear reactor spatial kinetics, are presented and compared. Both these methods allow developing neutronics description by means of a set of ordinary differential equations. The comparison of the outcomes provided by the two approaches focuses on the capability of evaluating the reactivity and the neutron flux shape in different reactor configurations, with reference to a TRIGA Mark II reactor. The results given by the POD-based approach are higher-fidelity with respect to the reference solution than those computed according to the MM-based approach, in particular when the perturbation concerns a reduced region of the core. If the perturbation is homogeneous throughout the core, the two approaches allow obtaining comparable accuracy results on the quantities of interest. As far as the computational burden is concerned, the POD approach ensures a better efficiency rather than direct Modal Method, thanks to the ability of performing a longer computation in the preprocessing that leads to a faster evaluation during the on-line phase.

@ARTICLE{SartoriBaroliCammiChiesaLuzziPonciroliPrevitaliRicottiRozzaSisti2014,
author = {Sartori, A. and Baroli, D. and Cammi, A. and Chiesa, D. and Luzzi,
L. and Ponciroli, R. and Previtali, E. and Ricotti, M.E. and Rozza,
G. and Sisti, M.},
title = {Comparison of a Modal Method and a Proper Orthogonal Decomposition
approach for multi-group time-dependent reactor spatial kinetics},
journal = {Annals of Nuclear Energy},
year = {2014},
volume = {71},
pages = {217--229},
abstract = {In this paper, two modelling approaches based on a Modal Method (MM)
and on the Proper Orthogonal Decomposition (POD) technique, for developing
a control-oriented model of nuclear reactor spatial kinetics, are
presented and compared. Both these methods allow developing neutronics
description by means of a set of ordinary differential equations.
The comparison of the outcomes provided by the two approaches focuses
on the capability of evaluating the reactivity and the neutron flux
shape in different reactor configurations, with reference to a TRIGA
Mark II reactor. The results given by the POD-based approach are
higher-fidelity with respect to the reference solution than those
computed according to the MM-based approach, in particular when the
perturbation concerns a reduced region of the core. If the perturbation
is homogeneous throughout the core, the two approaches allow obtaining
comparable accuracy results on the quantities of interest. As far
as the computational burden is concerned, the POD approach ensures
a better efficiency rather than direct Modal Method, thanks to the
ability of performing a longer computation in the preprocessing that
leads to a faster evaluation during the on-line phase.},
doi = {10.1016/j.anucene.2014.03.043}
}

14. A. Sartori, D. Baroli, A. Cammi, L. Luzzi, and G. Rozza, "A reduced order model for multi-group time-dependent parametrized reactor spatial kinetics", in International Conference on Nuclear Engineering, Proceedings, ICONE, 2014.
[BibTeX] [Abstract] [View on publisher website]
In this work, a Reduced Order Model (ROM) for multigroup time-dependent parametrized reactor spatial kinetics is presented. The Reduced Basis method (built upon a high-fidelity truth'' finite element approximation) has been applied to model the neutronics behavior of a parametrized system composed by a control rod surrounded by fissile material. The neutron kinetics has been described by means of a parametrized multi-group diffusion equation where the height of the control rod (i.e., how much the rod is inserted) plays the role of the varying parameter. In order to model a continuous movement of the rod, a piecewise affine transformation based on subdomain division has been implemented. The proposed ROM is capable to efficiently reproduce the neutron flux distribution allowing to take into account the spatial effects induced by the movement of the control rod with a computational speed-up of 30000 times, with respect to the truth'' model.

@INPROCEEDINGS{SartoriBaroliCammiLuzziRozza2014,
author = {Sartori, A. and Baroli, D. and Cammi, A. and Luzzi, L. and Rozza,
G.},
title = {A reduced order model for multi-group time-dependent parametrized
reactor spatial kinetics},
year = {2014},
volume = {5},
abstract = {In this work, a Reduced Order Model (ROM) for multigroup time-dependent
parametrized reactor spatial kinetics is presented. The Reduced Basis
method (built upon a high-fidelity truth'' finite element approximation)
has been applied to model the neutronics behavior of a parametrized
system composed by a control rod surrounded by fissile material.
The neutron kinetics has been described by means of a parametrized
multi-group diffusion equation where the height of the control rod
(i.e., how much the rod is inserted) plays the role of the varying
parameter. In order to model a continuous movement of the rod, a
piecewise affine transformation based on subdomain division has been
implemented. The proposed ROM is capable to efficiently reproduce
the neutron flux distribution allowing to take into account the spatial
effects induced by the movement of the control rod with a computational
speed-up of 30000 times, with respect to the truth'' model.},
doi = {10.1115/ICONE22-30698},
booktitle = {International Conference on Nuclear Engineering, Proceedings, ICONE}
}

### 2013

1. P. Chen, A. Quarteroni, and G. Rozza, "Stochastic optimal robin boundary control problems of advection-dominated elliptic equations", SIAM Journal on Numerical Analysis, 51(5), pp. 2700–2722, 2013.
In this work we deal with a stochastic optimal Robin boundary control problem constrained by an advection-diffusion-reaction elliptic equation with advection-dominated term. We assume that the uncertainty comes from the advection field and consider a stochastic Robin boundary condition as control function. A stochastic saddle point system is formulated and proved to be equivalent to the first order optimality system for the optimal control problem, based on which we provide the existence and uniqueness of the optimal solution as well as some results on stochastic regularity with respect to the random variables. Stabilized finite element approximations in physical space and collocation approximations in stochastic space are applied to discretize the optimality system. A global error estimate in the product of physical space and stochastic space for the numerical approximation is derived. Illustrative numerical experiments are provided.

@ARTICLE{ChenQuarteroniRozza2013,
author = {Chen, P. and Quarteroni, A. and Rozza, G.},
title = {Stochastic optimal robin boundary control problems of advection-dominated
elliptic equations},
journal = {SIAM Journal on Numerical Analysis},
year = {2013},
volume = {51},
pages = {2700--2722},
number = {5},
abstract = {In this work we deal with a stochastic optimal Robin boundary control
problem constrained by an advection-diffusion-reaction elliptic equation
with advection-dominated term. We assume that the uncertainty comes
from the advection field and consider a stochastic Robin boundary
condition as control function. A stochastic saddle point system is
formulated and proved to be equivalent to the first order optimality
system for the optimal control problem, based on which we provide
the existence and uniqueness of the optimal solution as well as some
results on stochastic regularity with respect to the random variables.
Stabilized finite element approximations in physical space and collocation
approximations in stochastic space are applied to discretize the
optimality system. A global error estimate in the product of physical
space and stochastic space for the numerical approximation is derived.
Illustrative numerical experiments are provided.},
doi = {10.1137/120884158},
preprint = {http://mathicse.epfl.ch/files/content/sites/mathicse/files/Mathicse%20reports%202012/23.2012_NEW_PC-AQ-GR.pdf}
}

2. P. Chen, A. Quarteroni, and G. Rozza, "A weighted reduced basis method for elliptic partial differential equations with random input data", SIAM Journal on Numerical Analysis, 51(6), pp. 3163–3185, 2013.
In this work we propose and analyze a weighted reduced basis method to solve elliptic partial differential equations (PDEs) with random input data. The PDEs are first transformed into a weighted parametric elliptic problem depending on a finite number of parameters. Distinctive importance of the solution at different values of the parameters is taken into account by assigning different weights to the samples in the greedy sampling procedure. A priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters. Numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and the stochastic collocation method in both univariate and multivariate stochastic problems.

@ARTICLE{ChenQuarteroniRozza2013a,
author = {Chen, P. and Quarteroni, A. and Rozza, G.},
title = {A weighted reduced basis method for elliptic partial differential
equations with random input data},
journal = {SIAM Journal on Numerical Analysis},
year = {2013},
volume = {51},
pages = {3163--3185},
number = {6},
abstract = {In this work we propose and analyze a weighted reduced basis method
to solve elliptic partial differential equations (PDEs) with random
input data. The PDEs are first transformed into a weighted parametric
elliptic problem depending on a finite number of parameters. Distinctive
importance of the solution at different values of the parameters
is taken into account by assigning different weights to the samples
in the greedy sampling procedure. A priori convergence analysis is
carried out by constructive approximation of the exact solution with
respect to the weighted parameters. Numerical examples are provided
for the assessment of the advantages of the proposed method over
the reduced basis method and the stochastic collocation method in
both univariate and multivariate stochastic problems.},
doi = {10.1137/130905253},
preprint = {http://mathicse.epfl.ch/files/content/sites/mathicse/files/Mathicse%20reports%202013/04.2013_NEW_PC-AQ-GR.pdf}
}

3. P. Chen, A. Quarteroni, and G. Rozza, "Simulation-based uncertainty quantification of human arterial network hemodynamics", International Journal for Numerical Methods in Biomedical Engineering, 29(6), pp. 698–721, 2013.
This work aims at identifying and quantifying uncertainties from various sources in human cardiovascular system based on stochastic simulation of a one-dimensional arterial network. A general analysis of different uncertainties and probability characterization with log-normal distribution of these uncertainties is introduced. Deriving from a deterministic one-dimensional fluid-structure interaction model, we establish the stochastic model as a coupled hyperbolic system incorporated with parametric uncertainties to describe the blood flow and pressure wave propagation in the arterial network. By applying a stochastic collocation method with sparse grid technique, we study systemically the statistics and sensitivity of the solution with respect to many different uncertainties in a relatively complete arterial network with potential physiological and pathological implications for the first time.

@ARTICLE{ChenQuarteroniRozza2013b,
author = {Chen, P. and Quarteroni, A. and Rozza, G.},
title = {Simulation-based uncertainty quantification of human arterial network
hemodynamics},
journal = {International Journal for Numerical Methods in Biomedical Engineering},
year = {2013},
volume = {29},
pages = {698--721},
number = {6},
abstract = {This work aims at identifying and quantifying uncertainties from various
sources in human cardiovascular system based on stochastic simulation
of a one-dimensional arterial network. A general analysis of different
uncertainties and probability characterization with log-normal distribution
of these uncertainties is introduced. Deriving from a deterministic
one-dimensional fluid-structure interaction model, we establish the
stochastic model as a coupled hyperbolic system incorporated with
parametric uncertainties to describe the blood flow and pressure
wave propagation in the arterial network. By applying a stochastic
collocation method with sparse grid technique, we study systemically
the statistics and sensitivity of the solution with respect to many
different uncertainties in a relatively complete arterial network
with potential physiological and pathological implications for the
first time.},
doi = {10.1002/cnm.2554},
preprint = {http://preprints.sissa.it/xmlui/bitstream/handle/1963/6592/report.pdf?sequence=1&isAllowed=y}
}

4. D. Devaud, A. Manzoni, and G. Rozza, "A combination between the reduced basis method and the ANOVA expansion: On the computation of sensitivity indices", Comptes Rendus Mathematique, 351(15-16), pp. 593–598, 2013.
We consider a method to efficiently evaluate in a real-time context an output based on the numerical solution of a partial differential equation depending on a large number of parameters. We state a result allowing to improve the computational performance of a three-step RB-ANOVA-RB method. This is a combination of the reduced basis (RB) method and the analysis of variations (ANOVA) expansion, aiming at compressing the parameter space without affecting the accuracy of the output. The idea of this method is to compute a first (coarse) RB approximation of the output of interest involving all the parameter components, but with a large tolerance on the a posteriori error estimate; then, we evaluate the ANOVA expansion of the output and freeze the least important parameter components; finally, considering a restricted model involving just the retained parameter components, we compute a second (fine) RB approximation with a smaller tolerance on the a posteriori error estimate. The fine RB approximation entails lower computational costs than the coarse one, because of the reduction of parameter dimensionality. Our result provides a criterion to avoid the computation of those terms in the ANOVA expansion that are related to the interaction between parameters in the bilinear form, thus making the RB-ANOVA-RB procedure computationally more feasible.

@ARTICLE{DevaudManzoniRozza2013,
author = {Devaud, D. and Manzoni, A. and Rozza, G.},
title = {A combination between the reduced basis method and the {ANOVA} expansion:
On the computation of sensitivity indices},
journal = {Comptes Rendus Mathematique},
year = {2013},
volume = {351},
pages = {593--598},
number = {15-16},
abstract = {We consider a method to efficiently evaluate in a real-time context
an output based on the numerical solution of a partial differential
equation depending on a large number of parameters. We state a result
allowing to improve the computational performance of a three-step
RB-ANOVA-RB method. This is a combination of the reduced basis (RB)
method and the analysis of variations (ANOVA) expansion, aiming at
compressing the parameter space without affecting the accuracy of
the output. The idea of this method is to compute a first (coarse)
RB approximation of the output of interest involving all the parameter
components, but with a large tolerance on the a posteriori error
estimate; then, we evaluate the ANOVA expansion of the output and
freeze the least important parameter components; finally, considering
a restricted model involving just the retained parameter components,
we compute a second (fine) RB approximation with a smaller tolerance
on the a posteriori error estimate. The fine RB approximation entails
lower computational costs than the coarse one, because of the reduction
of parameter dimensionality. Our result provides a criterion to avoid
the computation of those terms in the ANOVA expansion that are related
to the interaction between parameters in the bilinear form, thus
making the RB-ANOVA-RB procedure computationally more feasible.},
doi = {10.1016/j.crma.2013.07.023},
preprint = {http://preprints.sissa.it/xmlui/bitstream/handle/1963/7389/Devaud_Manzoni_Rozza_2013.pdf?sequence=1&isAllowed=y}
}

5. A. Koshakji, A. Quarteroni, and G. Rozza, "Free Form Deformation Techniques Applied to 3D Shape Optimization Problems", Communications in Applied and Industrial Mathematics, 2013.
[BibTeX] [Abstract] [View on publisher website]
The purpose of this work is to analyse and study an efficient parametrization technique for a 3D shape optimization problem. After a brief review of the techniques and approaches already available in literature, we recall the Free Form Deformation parametrization, a technique which proved to be efficient and at the same time versatile, allowing to manage complex shapes even with few parameters. We tested and studied the FFD technique by establishing a path, from the geometry definition, to the method implementation, and finally to the simulation and to the optimization of the shape. In particular, we have studied a bulb and a rudder of a race sailing boat as model applications, where we have tested a complete procedure from Computer-Aided-Design to build the geometrical model to discretization and mesh generation.

@ARTICLE{KoshakjiQuarteroniRozza2013,
author = {Anwar Koshakji and Alfio Quarteroni and Gianluigi Rozza},
title = {Free Form Deformation Techniques Applied to 3D Shape Optimization
Problems},
journal = {Communications in Applied and Industrial Mathematics},
year = {2013},
abstract = {The purpose of this work is to analyse and study an efficient parametrization
technique for a 3D shape optimization problem. After a brief review
of the techniques and approaches already available in literature,
we recall the Free Form Deformation parametrization, a technique
which proved to be efficient and at the same time versatile, allowing
to manage complex shapes even with few parameters. We tested and
studied the FFD technique by establishing a path, from the geometry
definition, to the method implementation, and finally to the simulation
and to the optimization of the shape. In particular, we have studied
a bulb and a rudder of a race sailing boat as model applications,
where we have tested a complete procedure from Computer-Aided-Design
to build the geometrical model to discretization and mesh generation.},
doi = {10.1685/journal.caim.452},
}

6. T. Lassila, A. Manzoni, A. Quarteroni, and G. Rozza, "Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty", ESAIM: Mathematical Modelling and Numerical Analysis, 47(4), pp. 1107–1131, 2013.
We review the optimal design of an arterial bypass graft following either a (i) boundary optimal control approach, or a (ii) shape optimization formulation. The main focus is quantifying and treating the uncertainty in the residual flow when the hosting artery is not completely occluded, for which the worst-case in terms of recirculation effects is inferred to correspond to a strong orifice flow through near-complete occlusion. A worst-case optimal control approach is applied to the steady Navier-Stokes equations in 2D to identify an anastomosis angle and a cuffed shape that are robust with respect to a possible range of residual flows. We also consider a reduced order modelling framework based on reduced basis methods in order to make the robust design problem computationally feasible. The results obtained in 2D are compared with simulations in a 3D geometry but without model reduction or the robust framework.

@ARTICLE{LassilaManzoniQuarteroniRozza2013,
author = {Lassila, T. and Manzoni, A. and Quarteroni, A. and Rozza, G.},
title = {Boundary control and shape optimization for the robust design of
bypass anastomoses under uncertainty},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
year = {2013},
volume = {47},
pages = {1107--1131},
number = {4},
abstract = {We review the optimal design of an arterial bypass graft following
either a (i) boundary optimal control approach, or a (ii) shape optimization
formulation. The main focus is quantifying and treating the uncertainty
in the residual flow when the hosting artery is not completely occluded,
for which the worst-case in terms of recirculation effects is inferred
to correspond to a strong orifice flow through near-complete occlusion.
A worst-case optimal control approach is applied to the steady Navier-Stokes
equations in 2D to identify an anastomosis angle and a cuffed shape
that are robust with respect to a possible range of residual flows.
We also consider a reduced order modelling framework based on reduced
basis methods in order to make the robust design problem computationally
feasible. The results obtained in 2D are compared with simulations
in a 3D geometry but without model reduction or the robust framework.},
doi = {10.1051/m2an/2012059},
preprint = {http://preprints.sissa.it/xmlui/bitstream/handle/1963/6337/LMQR_M2AN_Special_SISSAreport.pdf?sequence=1&isAllowed=y}
}

7. T. Lassila, A. Manzoni, A. Quarteroni, and G. Rozza, "A reduced computational and geometrical framework for inverse problems in hemodynamics", International Journal for Numerical Methods in Biomedical Engineering, 29(7), pp. 741–776, 2013.
The solution of inverse problems in cardiovascular mathematics is computationally expensive. In this paper, we apply a domain parametrization technique to reduce both the geometrical and computational complexities of the forward problem and replace the finite element solution of the incompressible Navier-Stokes equations by a computationally less-expensive reduced-basis approximation. This greatly reduces the cost of simulating the forward problem. We then consider the solution of inverse problems both in the deterministic sense, by solving a least-squares problem, and in the statistical sense, by using a Bayesian framework for quantifying uncertainty. Two inverse problems arising in hemodynamics modeling are considered: (i) a simplified fluid-structure interaction model problem in a portion of a stenosed artery for quantifying the risk of atherosclerosis by identifying the material parameters of the arterial wall on the basis of pressure measurements; (ii) a simplified femoral bypass graft model for robust shape design under uncertain residual flow in the main arterial branch identified from pressure measurements.

@ARTICLE{LassilaManzoniQuarteroniRozza2013a,
author = {Lassila, T. and Manzoni, A. and Quarteroni, A. and Rozza, G.},
title = {A reduced computational and geometrical framework for inverse problems
in hemodynamics},
journal = {International Journal for Numerical Methods in Biomedical Engineering},
year = {2013},
volume = {29},
pages = {741--776},
number = {7},
abstract = {The solution of inverse problems in cardiovascular mathematics is
computationally expensive. In this paper, we apply a domain parametrization
technique to reduce both the geometrical and computational complexities
of the forward problem and replace the finite element solution of
the incompressible Navier-Stokes equations by a computationally less-expensive
reduced-basis approximation. This greatly reduces the cost of simulating
the forward problem. We then consider the solution of inverse problems
both in the deterministic sense, by solving a least-squares problem,
and in the statistical sense, by using a Bayesian framework for quantifying
uncertainty. Two inverse problems arising in hemodynamics modeling
are considered: (i) a simplified fluid-structure interaction model
problem in a portion of a stenosed artery for quantifying the risk
of atherosclerosis by identifying the material parameters of the
arterial wall on the basis of pressure measurements; (ii) a simplified
femoral bypass graft model for robust shape design under uncertain
residual flow in the main arterial branch identified from pressure
measurements.},
doi = {10.1002/cnm.2559},
preprint = {http://preprints.sissa.it/xmlui/bitstream/handle/1963/6971/LMQR_inverse_problems_Haemo.pdf?sequence=1&isAllowed=y}
}

8. T. Lassila, A. Manzoni, A. Quarteroni, and G. Rozza, "Generalized reduced basis methods and n-width estimates for the approximation of the solution manifold of parametric PDEs", in Analysis and Numerics of Partial Differential Equations, F. Brezzi, P. Colli Franzone, U. Gianazza, and G. Gilardi (eds.), , vol. 4, pp. 307–329, 2013.
The set of solutions of a parameter-dependent linear partial differential equation with smooth coefficients typically forms a compact manifold in a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold We focus on operators showing an affine parametric dependence, expressed as a linear combination of parameter-independent operators through some smooth, parameter-dependent scalar functions. In the case that the parameter-dependent operator has a dominant term in its affine expansion, one can prove the existence of exponentially convergent uniform approximation spaces for the entire solution manifold These spaces can be constructed without any assumptions on the parametric regularity of the manifold - only spatial regularity of the solutions is required The exponential convergence rate is then inherited by the generalized reduced basis method We provide a numerical example related to parametrized elliptic equations confirming the predicted convergence rates.

@INBOOK{LassilaManzoniQuarteroniRozza2013b,
author = {Lassila, T. and Manzoni, A. and Quarteroni, A. and Rozza, G.},
chapter = {Generalized reduced basis methods and n-width estimates for the approximation
of the solution manifold of parametric PDEs},
editor={Brezzi, Franco and Colli Franzone, Piero and Gianazza, Ugo and Gilardi, Gianni},
booktitle={Analysis and Numerics of Partial Differential Equations},
year = {2013},
volume = {4},
pages = {307--329},
abstract = {The set of solutions of a parameter-dependent linear partial differential
equation with smooth coefficients typically forms a compact manifold
in a Hilbert space. In this paper we review the generalized reduced
basis method as a fast computational tool for the uniform approximation
of the solution manifold We focus on operators showing an affine
parametric dependence, expressed as a linear combination of parameter-independent
operators through some smooth, parameter-dependent scalar functions.
In the case that the parameter-dependent operator has a dominant
term in its affine expansion, one can prove the existence of exponentially
convergent uniform approximation spaces for the entire solution manifold
These spaces can be constructed without any assumptions on the parametric
regularity of the manifold - only spatial regularity of the solutions
is required The exponential convergence rate is then inherited by
the generalized reduced basis method We provide a numerical example
related to parametrized elliptic equations confirming the predicted
convergence rates.},
doi={10.1007/978-88-470-2592-9_16},
preprint = {http://preprints.sissa.it/xmlui/bitstream/handle/1963/6340/qlmr-bumi_FINAL_SISSAreport.pdf?sequence=1&isAllowed=y}
}

9. T. Lassila, A. Manzoni, and G. Rozza, "Reduction strategies for shape dependent inverse problems in haemodynamics", IFIP Advances in Information and Communication Technology, 391 AICT, pp. 397–406, 2013.
This work deals with the development and application of reduction strategies for real-time and many query problems arising in fluid dynamics, such as shape optimization, shape registration (reconstruction), and shape parametrization. The proposed strategy is based on the coupling between reduced basis methods for the reduction of computational complexity and suitable shape parametrizations - such as free-form deformations or radial basis functions - for low-dimensional geometrical description. Our focus is on problems arising in haemodynamics: efficient shape parametrization of cardiovascular geometries (e.g. bypass grafts, carotid artery bifurcation, stenosed artery sections) for the rapid blood flow simulation - and related output evaluation - in domains of variable shape (e.g. vessels in presence of growing stenosis) provide an example of a class of problems which can be recast in the real-time or in the many-query context.

@ARTICLE{LassilaManzoniRozza2013,
author = {Lassila, T. and Manzoni, A. and Rozza, G.},
title = {Reduction strategies for shape dependent inverse problems in haemodynamics},
journal = {IFIP Advances in Information and Communication Technology},
year = {2013},
volume = {391 AICT},
pages = {397--406},
abstract = {This work deals with the development and application of reduction
strategies for real-time and many query problems arising in fluid
dynamics, such as shape optimization, shape registration (reconstruction),
and shape parametrization. The proposed strategy is based on the
coupling between reduced basis methods for the reduction of computational
complexity and suitable shape parametrizations - such as free-form
deformations or radial basis functions - for low-dimensional geometrical
description. Our focus is on problems arising in haemodynamics: efficient
shape parametrization of cardiovascular geometries (e.g. bypass grafts,
carotid artery bifurcation, stenosed artery sections) for the rapid
blood flow simulation - and related output evaluation - in domains
of variable shape (e.g. vessels in presence of growing stenosis)
provide an example of a class of problems which can be recast in
the real-time or in the many-query context.},
doi = {10.1007/978-3-642-36062-6_40},
preprint = {http://mathicse.epfl.ch/files/content/sites/mathicse/files/Mathicse%20reports%202012/09.2012_TL-AM-GR.pdf}
}

10. F. Negri, G. Rozza, A. Manzoni, and A. Quarteroni, "Reduced basis method for parametrized elliptic optimal control problems", SIAM Journal on Scientific Computing, 35(5), pp. A2316–A2340, 2013.
We propose a suitable model reduction paradigm-the certified reduced basis method (RB)-for the rapid and reliable solution of parametrized optimal control problems governed by partial differential equations. In particular, we develop the methodology for parametrized quadratic optimization problems with elliptic equations as a constraint and infinite-dimensional control variable. First, we recast the optimal control problem in the framework of saddle-point problems in order to take advantage of the already developed RB theory for Stokes-type problems. Then, the usual ingredients of the RB methodology are called into play: a Galerkin projection onto a low-dimensional space of basis functions properly selected by an adaptive procedure; an affine parametric dependence enabling one to perform competitive offline-online splitting in the computational procedure; and an efficient and rigorous a posteriori error estimate on the state, control, and adjoint variables as well as on the cost functional. Finally, we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique.

@ARTICLE{NegriRozzaManzoniQuarteroni2013,
author = {Negri, F. and Rozza, G. and Manzoni, A. and Quarteroni, A.},
title = {Reduced basis method for parametrized elliptic optimal control problems},
journal = {SIAM Journal on Scientific Computing},
year = {2013},
volume = {35},
pages = {A2316--A2340},
number = {5},
abstract = {We propose a suitable model reduction paradigm-the certified reduced
basis method (RB)-for the rapid and reliable solution of parametrized
optimal control problems governed by partial differential equations.
In particular, we develop the methodology for parametrized quadratic
optimization problems with elliptic equations as a constraint and
infinite-dimensional control variable. First, we recast the optimal
control problem in the framework of saddle-point problems in order
problems. Then, the usual ingredients of the RB methodology are called
into play: a Galerkin projection onto a low-dimensional space of
basis functions properly selected by an adaptive procedure; an affine
parametric dependence enabling one to perform competitive offline-online
splitting in the computational procedure; and an efficient and rigorous
a posteriori error estimate on the state, control, and adjoint variables
as well as on the cost functional. Finally, we address some numerical
tests that confirm our theoretical results and show the efficiency
of the proposed technique.},
doi = {10.1137/120894737},
preprint = {http://mathicse.epfl.ch/files/content/sites/mathicse/files/Mathicse%20reports%202012/40.2012_FN-GR-AM-AQ.pdf}
}

11. G. Rozza, D. B. P. Huynh, and A. Manzoni, "Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: Roles of the inf-sup stability constants", Numerische Mathematik, 125(1), pp. 115–152, 2013.
In this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady Stokes flows in affinely parametrized geometries, focusing on the role played by the Brezzi's and Babuška's stability constants. The crucial ingredients of the methodology are a Galerkin projection onto a low-dimensional space of basis functions properly selected, an affine parametric dependence enabling to perform competitive Offline-Online splitting in the computational procedure and a rigorous a posteriori error estimation on field variables. The combinatiofn of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction, ideally suited for real-time simulation and many-query contexts (e.g. optimization, control or parameter identification). In particular, in this work we focus on (i) the stability of the reduced basis approximation based on the Brezzi's saddle point theory and the introduction of a supremizer operator on the pressure terms, (ii) a rigorous a posteriori error estimation procedure for velocity and pressure fields based on the Babuška's inf-sup constant (including residuals calculations), (iii) the computation of a lower bound of the stability constant, and (iv) different options for the reduced basis spaces construction. We present some illustrative results for both interior and external steady Stokes flows in parametrized geometries representing two parametrized classical Poiseuille and Couette flows, a channel contraction and a simple flow control problem around a curved obstacle.

@ARTICLE{RozzaHuynhManzoni2013,
author = {Rozza, G. and Huynh, D.B.P. and Manzoni, A.},
title = {Reduced basis approximation and a posteriori error estimation for
{S}tokes flows in parametrized geometries: Roles of the inf-sup stability
constants},
journal = {Numerische Mathematik},
year = {2013},
volume = {125},
pages = {115--152},
number = {1},
abstract = {In this paper we review and we extend the reduced basis approximation
and a posteriori error estimation for steady Stokes flows in affinely
parametrized geometries, focusing on the role played by the Brezzi's
and Babu\v{s}ka's stability constants. The crucial ingredients of
the methodology are a Galerkin projection onto a low-dimensional
space of basis functions properly selected, an affine parametric
dependence enabling to perform competitive Offline-Online splitting
in the computational procedure and a rigorous a posteriori error
estimation on field variables. The combinatiofn of these three factors
yields substantial computational savings which are at the basis of
an efficient model order reduction, ideally suited for real-time
simulation and many-query contexts (e.g. optimization, control or
parameter identification). In particular, in this work we focus on
(i) the stability of the reduced basis approximation based on the
Brezzi's saddle point theory and the introduction of a supremizer
operator on the pressure terms, (ii) a rigorous a posteriori error
estimation procedure for velocity and pressure fields based on the
Babu\v{s}ka's inf-sup constant (including residuals calculations),
(iii) the computation of a lower bound of the stability constant,
and (iv) different options for the reduced basis spaces construction.
We present some illustrative results for both interior and external
steady Stokes flows in parametrized geometries representing two parametrized
classical Poiseuille and Couette flows, a channel contraction and
a simple flow control problem around a curved obstacle.},
doi = {10.1007/s00211-013-0534-8},
preprint = {http://preprints.sissa.it/xmlui/bitstream/handle/1963/6339/Stokes_HMR10_SISSA_report.pdf?sequence=1&isAllowed=y}
}

### 2012

1. D. Ambrosi, A. Quarteroni, and G. Rozza, Modeling of Physiological Flows), Springer Milan, 2012, vol. 5.
[BibTeX] [View on publisher website]
@BOOK{AmbrosiQuarteroniRozza2012,
title = {{M}odeling of {P}hysiological {F}lows},
year = {2012},
author = {Ambrosi, D. and Quarteroni, A. and Rozza, G.},
volume = {5},
series = {Modeling, Simulation and Applications},
publisher = {Springer Milan},
doi = {10.1007/978-88-470-1935-5}
}

2. L. Iapichino, A. Quarteroni, and G. Rozza, "A reduced basis hybrid method for the coupling of parametrized domains represented by fluidic networks", Computer Methods in Applied Mechanics and Engineering, 221-222, pp. 63–82, 2012.
In this paper we propose a reduced basis hybrid method (RBHM) for the approximation of partial differential equations in domains represented by complex networks where topological features are recurrent. The RBHM is applied to Stokes equations in domains which are decomposable into smaller similar blocks that are properly coupled.The RBHM is built upon the reduced basis element method (RBEM) and it takes advantage from both the reduced basis methods (RB) and the domain decomposition method. We move from the consideration that the blocks composing the computational domain are topologically similar to a few reference shapes. On the latter, representative solutions, corresponding to the same governing partial differential equations, are computed for different values of some parameters of interest, representing, for example, the deformation of the blocks. A generalized transfinite mapping is used in order to produce a global map from the reference shapes of each block to any deformed configuration.The desired solution on the given original computational domain is recovered as projection of the previously precomputed solutions and then glued across subdomain interfaces by suitable coupling conditions.The geometrical parametrization of the domain, by transfinite mapping, induces non-affine parameter dependence: an empirical interpolation technique is used to recover an approximate affine parameter dependence and a subsequent offline/online decomposition of the reduced basis procedure. This computational decomposition yields a considerable reduction of the problem complexity. Results computed on some combinations of 2D and 3D geometries representing cardiovascular networks show the advantage of the method in terms of reduced computational costs and the quality of the coupling to guarantee continuity of both stresses, pressure and velocity at subdomain interfaces.

@ARTICLE{IapichinoQuarteroniRozza2012,
author = {Iapichino, L. and Quarteroni, A. and Rozza, G.},
title = {A reduced basis hybrid method for the coupling of parametrized domains
represented by fluidic networks},
journal = {Computer Methods in Applied Mechanics and Engineering},
year = {2012},
volume = {221-222},
pages = {63--82},
abstract = {In this paper we propose a reduced basis hybrid method (RBHM) for
the approximation of partial differential equations in domains represented
by complex networks where topological features are recurrent. The
RBHM is applied to Stokes equations in domains which are decomposable
into smaller similar blocks that are properly coupled.The RBHM is
built upon the reduced basis element method (RBEM) and it takes advantage
from both the reduced basis methods (RB) and the domain decomposition
method. We move from the consideration that the blocks composing
the computational domain are topologically similar to a few reference
shapes. On the latter, representative solutions, corresponding to
the same governing partial differential equations, are computed for
different values of some parameters of interest, representing, for
example, the deformation of the blocks. A generalized transfinite
mapping is used in order to produce a global map from the reference
shapes of each block to any deformed configuration.The desired solution
on the given original computational domain is recovered as projection
of the previously precomputed solutions and then glued across subdomain
interfaces by suitable coupling conditions.The geometrical parametrization
of the domain, by transfinite mapping, induces non-affine parameter
dependence: an empirical interpolation technique is used to recover
an approximate affine parameter dependence and a subsequent offline/online
decomposition of the reduced basis procedure. This computational
decomposition yields a considerable reduction of the problem complexity.
Results computed on some combinations of 2D and 3D geometries representing
cardiovascular networks show the advantage of the method in terms
of reduced computational costs and the quality of the coupling to
guarantee continuity of both stresses, pressure and velocity at subdomain
interfaces.},
doi = {10.1016/j.cma.2012.02.005},
preprint = {https://infoscience.epfl.ch/record/175878/files/RBHM_IQR_R2.pdf}
}

3. T. Lassila, A. Manzoni, and G. Rozza, "On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition", ESAIM: Mathematical Modelling and Numerical Analysis, 46(6), pp. 1555–1576, 2012.
A new approach for computationally efficient estimation of stability factors for parametric partial differential equations is presented. The general parametric bilinear form of the problem is approximated by two affinely parametrized bilinear forms at different levels of accuracy (after an empirical interpolation procedure). The successive constraint method is applied on the coarse level to obtain a lower bound for the stability factors, and this bound is extended to the fine level by adding a proper correction term. Because the approximate problems are affine, an efficient offline/online computational scheme can be developed for the certified solution (error bounds and stability factors) of the parametric equations considered. We experiment with different correction terms suited for a posteriori error estimation of the reduced basis solution of elliptic coercive and noncoercive problems.

@ARTICLE{LassilaManzoniRozza2012,
author = {Lassila, T. and Manzoni, A. and Rozza, G.},
title = {On the approximation of stability factors for general parametrized
partial differential equations with a two-level affine decomposition},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
year = {2012},
volume = {46},
pages = {1555--1576},
number = {6},
abstract = {A new approach for computationally efficient estimation of stability
factors for parametric partial differential equations is presented.
The general parametric bilinear form of the problem is approximated
by two affinely parametrized bilinear forms at different levels of
accuracy (after an empirical interpolation procedure). The successive
constraint method is applied on the coarse level to obtain a lower
bound for the stability factors, and this bound is extended to the
fine level by adding a proper correction term. Because the approximate
problems are affine, an efficient offline/online computational scheme
can be developed for the certified solution (error bounds and stability
factors) of the parametric equations considered. We experiment with
different correction terms suited for a posteriori error estimation
of the reduced basis solution of elliptic coercive and noncoercive
problems.},
doi = {10.1051/m2an/2012016},
preprint = {http://mathicse.epfl.ch/files/content/sites/mathicse/files/Mathicse%20reports%202011/08-2011%20TL-AM-GR.pdf}
}

4. T. Lassila, A. Quarteroni, and G. Rozza, "A reduced basis model with parametric coupling for fluid-structure interaction problems", SIAM Journal on Scientific Computing, 34(2), pp. A1187–A1213, 2012.
We present a new model reduction technique for steady fluid-structure interaction problems. When the fluid domain deformation is suitably parametrized, the coupling conditions between the fluid and the structure can be formulated in the low-dimensional space of geometric parameters. Moreover, we apply the reduced basis method to reduce the cost of repeated fluid solutions necessary to achieve convergence of fluid-structure iterations. In this way a reduced order model with reliable a posteriori error bounds is obtained. The proposed method is validated with an example of steady Stokes flow in an axisymmetric channel, where the structure is described by a simple one-dimensional generalized string model. We demonstrate rapid convergence of the reduced solution of the parametrically coupled problem as the number of geometric parameters is increased.

@ARTICLE{LassilaQuarteroniRozza2012,
author = {Lassila, T. and Quarteroni, A. and Rozza, G.},
title = {A reduced basis model with parametric coupling for fluid-structure
interaction problems},
journal = {SIAM Journal on Scientific Computing},
year = {2012},
volume = {34},
pages = {A1187--A1213},
number = {2},
abstract = {We present a new model reduction technique for steady fluid-structure
interaction problems. When the fluid domain deformation is suitably
parametrized, the coupling conditions between the fluid and the structure
can be formulated in the low-dimensional space of geometric parameters.
Moreover, we apply the reduced basis method to reduce the cost of
repeated fluid solutions necessary to achieve convergence of fluid-structure
iterations. In this way a reduced order model with reliable a posteriori
error bounds is obtained. The proposed method is validated with an
example of steady Stokes flow in an axisymmetric channel, where the
structure is described by a simple one-dimensional generalized string
model. We demonstrate rapid convergence of the reduced solution of
the parametrically coupled problem as the number of geometric parameters
is increased.},
doi = {10.1137/110819950},
preprint = {http://mathicse.epfl.ch/files/content/sites/mathicse/files/Mathicse%20reports%202011/01.2011%20TL-AQ-GR.pdf}
}

5. M. Lombardi, N. Parolini, A. Quarteroni, and G. Rozza, "Numerical Simulation of Sailing Boats: Dynamics, FSI, and Shape Optimization", , G. Buttazzo and A. Frediani (eds.), Springer US, pp. 339–377, 2012.
@INBOOK{LombardiParoliniQuarteroniRozza2012,
chapter = {Numerical Simulation of Sailing Boats: Dynamics, FSI, and Shape Optimization},
pages = {339--377},
title = {Variational Analysis and Aerospace Engineering: Mathematical Challenges
for Aerospace Design: Contributions from a Workshop held at the School
of Mathematics in Erice, Italy},
publisher = {Springer US},
year = {2012},
editor = {Buttazzo, Giuseppe and Frediani, Aldo},
author = {Lombardi, Matteo and Parolini, Nicola and Quarteroni, Alfio and Rozza,
Gianluigi},
doi = {10.1007/978-1-4614-2435-2_15},
isbn = {978-1-4614-2435-2},
preprint = {https://infoscience.epfl.ch/record/175879/files/PaerErice-Lombardi-parolini-quarteroni-Rozza.pdf}
}

6. A. Manzoni, A. Quarteroni, and G. Rozza, "Shape optimization for viscous flows by reduced basis methods and free-form deformation", International Journal for Numerical Methods in Fluids, 70(5), pp. 646–670, 2012.
In this paper, we further develop an approach previously introduced in Lassila and Rozza, 2010, for shape optimization that combines a suitable low-dimensional parametrization of the geometry (yielding a geometrical reduction) with reduced basis methods (yielding a reduction of computational complexity). More precisely, free-form deformation techniques are considered for the geometry description and its parametrization, whereas reduced basis methods are used upon a FE discretization to solve systems of parametrized partial differential equations. This allows an efficient flow field computation and cost functional evaluation during the iterative optimization procedure, resulting in effective computational savings with respect to usual shape optimization strategies. This approach is very general and can be applied to a broad variety of problems. In this paper, we apply it to find the optimal shape of aorto-coronaric bypass anastomoses based on vorticity minimization in the down-field region. Blood flows in the coronary arteries are modeled using Stokes equations; afterwards, results have been verified in feedback using Navier-Stokes equations.

@ARTICLE{ManzoniQuarteroniRozza2012,
author = {Manzoni, A. and Quarteroni, A. and Rozza, G.},
title = {Shape optimization for viscous flows by reduced basis methods and
free-form deformation},
journal = {International Journal for Numerical Methods in Fluids},
year = {2012},
volume = {70},
pages = {646--670},
number = {5},
abstract = {In this paper, we further develop an approach previously introduced
in Lassila and Rozza, 2010, for shape optimization that combines
a suitable low-dimensional parametrization of the geometry (yielding
a geometrical reduction) with reduced basis methods (yielding a reduction
of computational complexity). More precisely, free-form deformation
techniques are considered for the geometry description and its parametrization,
whereas reduced basis methods are used upon a FE discretization to
solve systems of parametrized partial differential equations. This
allows an efficient flow field computation and cost functional evaluation
during the iterative optimization procedure, resulting in effective
computational savings with respect to usual shape optimization strategies.
This approach is very general and can be applied to a broad variety
of problems. In this paper, we apply it to find the optimal shape
of aorto-coronaric bypass anastomoses based on vorticity minimization
in the down-field region. Blood flows in the coronary arteries are
modeled using Stokes equations; afterwards, results have been verified
in feedback using Navier-Stokes equations.},
doi = {10.1002/fld.2712},
}

7. A. Manzoni, A. Quarteroni, and G. Rozza, "Computational Reduction for Parametrized PDEs: Strategies and Applications", Milan Journal of Mathematics, 80(2), pp. 283–309, 2012.
[BibTeX] [Abstract] [View on publisher website]
In this paper we present a compact review on the mostly used techniques for computational reduction in numerical approximation of partial differential equations. We highlight the common features of these techniques and provide a detailed presentation of the reduced basis method, focusing on greedy algorithms for the construction of the reduced spaces. An alternative family of reduction techniques based on surrogate response surface models is briefly recalled too. Then, a simple example dealing with inviscid flows is presented, showing the reliability of the reduced basis method and a comparison between this technique and some surrogate models.

@ARTICLE{ManzoniQuarteroniRozza2012a,
author = {Manzoni, A. and Quarteroni, A. and Rozza, G.},
title = {Computational Reduction for Parametrized {PDEs}: Strategies and Applications},
journal = {Milan Journal of Mathematics},
year = {2012},
volume = {80},
pages = {283--309},
number = {2},
abstract = {In this paper we present a compact review on the mostly used techniques
for computational reduction in numerical approximation of partial
differential equations. We highlight the common features of these
techniques and provide a detailed presentation of the reduced basis
method, focusing on greedy algorithms for the construction of the
reduced spaces. An alternative family of reduction techniques based
on surrogate response surface models is briefly recalled too. Then,
a simple example dealing with inviscid flows is presented, showing
the reliability of the reduced basis method and a comparison between
this technique and some surrogate models.},
doi = {10.1007/s00032-012-0182-y}
}

8. A. Manzoni, A. Quarteroni, and G. Rozza, "Model reduction techniques for fast blood flow simulation in parametrized geometries", International Journal for Numerical Methods in Biomedical Engineering, 28(6-7), pp. 604–625, 2012.
In this paper, we propose a new model reduction technique aimed at real-time blood flow simulations on a given family of geometrical shapes of arterial vessels. Our approach is based on the combination of a low-dimensional shape parametrization of the computational domain and the reduced basis method to solve the associated parametrized flow equations. We propose a preliminary analysis carried on a set of arterial vessel geometries, described by means of a radial basis functions parametrization. In order to account for patient-specific arterial configurations, we reconstruct the latter by solving a suitable parameter identification problem. Real-time simulation of blood flows are thus performed on each reconstructed parametrized geometry, by means of the reduced basis method. We focus on a family of parametrized carotid artery bifurcations, by modelling blood flows using Navier-Stokes equations and measuring distributed outputs such as viscous energy dissipation or vorticity. The latter are indexes that might be correlated with the assessment of pathological risks. The approach advocated here can be applied to a broad variety of (different) flow problems related with geometry/shape variation, for instance related with shape sensitivity analysis, parametric exploration and shape design.

@ARTICLE{ManzoniQuarteroniRozza2012b,
author = {Manzoni, A. and Quarteroni, A. and Rozza, G.},
title = {Model reduction techniques for fast blood flow simulation in parametrized
geometries},
journal = {International Journal for Numerical Methods in Biomedical Engineering},
year = {2012},
volume = {28},
pages = {604--625},
number = {6-7},
abstract = {In this paper, we propose a new model reduction technique aimed at
real-time blood flow simulations on a given family of geometrical
shapes of arterial vessels. Our approach is based on the combination
of a low-dimensional shape parametrization of the computational domain
and the reduced basis method to solve the associated parametrized
flow equations. We propose a preliminary analysis carried on a set
of arterial vessel geometries, described by means of a radial basis
functions parametrization. In order to account for patient-specific
arterial configurations, we reconstruct the latter by solving a suitable
parameter identification problem. Real-time simulation of blood flows
are thus performed on each reconstructed parametrized geometry, by
means of the reduced basis method. We focus on a family of parametrized
carotid artery bifurcations, by modelling blood flows using Navier-Stokes
equations and measuring distributed outputs such as viscous energy
dissipation or vorticity. The latter are indexes that might be correlated
with the assessment of pathological risks. The approach advocated
here can be applied to a broad variety of (different) flow problems
related with geometry/shape variation, for instance related with
shape sensitivity analysis, parametric exploration and shape design.},
doi = {10.1002/cnm.1465},
preprint = {https://infoscience.epfl.ch/record/167776/files/Manzoni_Quarteroni_Rozza_IJNMBE_moxreport.pdf}
}

9. G. Rozza, A. Manzoni, and F. Negri, "Reduction strategies for PDE-constrained optimization problems in haemodynamics", in ECCOMAS 2012 - European Congress on Computational Methods in Applied Sciences and Engineering, 2012, pp. 1749–1768.
Solving optimal control problems for many different scenarios obtained by varying a set of parameters in the state system is a computationally extensive task. In this paper we present a new reduced framework for the formulation, the analysis and the numerical solution of parametrized PDE-constrained optimization problems. This framework is based on a suitable saddle-point formulation of the optimal control problem and exploits the reduced basis method for the rapid and reliable solution of parametrized PDEs, leading to a relevant computational reduction with respect to traditional discretization techniques such as the finite element method. This allows a very efficient evaluation of state solutions and cost functionals, leading to an effective solution of repeated optimal control problems, even on domains of variable shape, for which a further (geometrical) reduction is pursued, relying on flexible shape parametrization techniques. This setting is applied to the solution of two problems arising from haemodynamics, dealing with both data reconstruction and data assimilation over domains of variable shape, which can be recast in a common PDE-constrained optimization formulation.

@INPROCEEDINGS{RozzaManzoniNegri2012,
author = {Rozza, G. and Manzoni, A. and Negri, F.},
title = {Reduction strategies for {PDE}-constrained optimization problems
in haemodynamics},
year = {2012},
pages = {1749--1768},
abstract = {Solving optimal control problems for many different scenarios obtained
by varying a set of parameters in the state system is a computationally
extensive task. In this paper we present a new reduced framework
for the formulation, the analysis and the numerical solution of parametrized
PDE-constrained optimization problems. This framework is based on
a suitable saddle-point formulation of the optimal control problem
and exploits the reduced basis method for the rapid and reliable
solution of parametrized PDEs, leading to a relevant computational
reduction with respect to traditional discretization techniques such
as the finite element method. This allows a very efficient evaluation
of state solutions and cost functionals, leading to an effective
solution of repeated optimal control problems, even on domains of
variable shape, for which a further (geometrical) reduction is pursued,
relying on flexible shape parametrization techniques. This setting
is applied to the solution of two problems arising from haemodynamics,
dealing with both data reconstruction and data assimilation over
domains of variable shape, which can be recast in a common PDE-constrained
optimization formulation.},
booktitle = {ECCOMAS 2012 - European Congress on Computational Methods in Applied
Sciences and Engineering},
preprint = {http://cmcs.epfl.ch/files/content/sites/mathicse/files/Mathicse%20reports%202012/26.2012_GR-AM-FN.pdf}
}

### 2011

1. F. Gelsomino and G. Rozza, "Comparison and combination of reduced-order modelling techniques in 3D parametrized heat transfer problems", Mathematical and Computer Modelling of Dynamical Systems, 17(4), pp. 371–394, 2011.
Reduced basis method has successfully been used in 2D to solve heat transfer parametrized problems. In this work, we present some 3D applications in the same field.We consider two problems, the steady Thermal Fin and the time-dependent Graetz Flow, we compare two reduced-order modelling techniques: Reduced basis and Proper orthogonal decomposition, then we apply a combination of the two strategies in the time-dependent case.

@ARTICLE{GelsominoRozza2011,
author = {Gelsomino, F. and Rozza, G.},
title = {Comparison and combination of reduced-order modelling techniques
in {3D} parametrized heat transfer problems},
journal = {Mathematical and Computer Modelling of Dynamical Systems},
year = {2011},
volume = {17},
pages = {371--394},
number = {4},
abstract = {Reduced basis method has successfully been used in 2D to solve heat
transfer parametrized problems. In this work, we present some 3D
applications in the same field.We consider two problems, the steady
Thermal Fin and the time-dependent Graetz Flow, we compare two reduced-order
modelling techniques: Reduced basis and Proper orthogonal decomposition,
then we apply a combination of the two strategies in the time-dependent
case.},
doi = {10.1080/13873954.2011.547672},
preprint = {https://infoscience.epfl.ch/record/163349/files/Rozza_gelsomino_MCMDS_rev.pdf}
}

2. T. Lassila and G. Rozza, "Model reduction of semiaffinely parameterized partial differential equations by two-level affine approximation", Comptes Rendus Mathematique, 349(1-2), pp. 61–66, 2011.
We propose an improvement to the reduced basis method for parametric partial differential equations. An assumption of affine parameterization leads to an efficient offline-online decomposition when the problem is solved for many different parametric configurations. We consider an advection-diffusion problem, where the diffusive term is non-affinely parameterized and treated with a two-level affine approximation given by the empirical interpolation method. The offline stage and a posteriori error estimation is performed using the coarse-level approximation, while the fine-level approximation is used to perform a correction iteration that reduces the actual error of the reduced basis approximation while keeping the same certified error bounds.

@ARTICLE{LassilaRozza2011,
author = {Lassila, T. and Rozza, G.},
title = {Model reduction of semiaffinely parameterized partial differential
equations by two-level affine approximation},
journal = {Comptes Rendus Mathematique},
year = {2011},
volume = {349},
pages = {61--66},
number = {1-2},
abstract = {We propose an improvement to the reduced basis method for parametric
partial differential equations. An assumption of affine parameterization
leads to an efficient offline-online decomposition when the problem
is solved for many different parametric configurations. We consider
an advection-diffusion problem, where the diffusive term is non-affinely
parameterized and treated with a two-level affine approximation given
by the empirical interpolation method. The offline stage and a posteriori
error estimation is performed using the coarse-level approximation,
while the fine-level approximation is used to perform a correction
iteration that reduces the actual error of the reduced basis approximation
while keeping the same certified error bounds.},
doi = {10.1016/j.crma.2010.11.016},
preprint = {https://infoscience.epfl.ch/record/155005/files/CRAS_Lassila_Rozza_Revised.pdf}
}

3. A. Quarteroni, A. Manzoni, and G. Rozza, "Model order reduction by reduced basis methods and free-form deformations for shape optimization", in Schnelle Löser für Partielle Differentialgleichungen, 2011, pp. 19–22.
[BibTeX] [View on publisher website]
@INPROCEEDINGS{QuarteroniManzoniRozza2011,
author = {Quarteroni, Alfio and Manzoni, Andrea and Rozza, Gianluigi},
title = {Model order reduction by reduced basis methods and free-form deformations
for shape optimization},
booktitle = {Schnelle {L}\"oser f\"ur {P}artielle {D}ifferentialgleichungen},
year = {2011},
editor = {Bank, Randolph and Hackbusch, Wolfgang and Wittum, Gabriel},
number = {28},
series = {Mathematisches Forschungsinstitut Oberwolfach},
pages = {19--22},
doi = {10.4171/OWR/2011/28},
}

4. A. Quarteroni, G. Rozza, and A. Manzoni, "Certified reduced basis approximation for parametrized partial differential equations and applications", Journal of Mathematics in Industry, 1(1), pp. 1–49, 2011.
Reduction strategies, such as model order reduction (MOR) or reduced basis (RB) methods, in scientific computing may become crucial in applications of increasing complexity. In this paper we review the reduced basis method (built upon a high-fidelity truth'' finite element approximation) for a rapid and reliable approximation of parametrized partial differential equations, and comment on their potential impact on applications of industrial interest. The essential ingredients of RB methodology are: a Galerkin projection onto a low-dimensional space of basis functions properly selected, an affine parametric dependence enabling to perform a competitive Offline-Online splitting in the computational procedure, and a rigorous a posteriori error estimation used for both the basis selection and the certification of the solution. The combination of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction, ideally suited for real-time simulation and many-query contexts (e.g. optimization, control or parameter identification). After a brief excursus on the methodology, we focus on linear elliptic and parabolic problems, discussing some extensions to more general classes of problems and several perspectives of the ongoing research. We present some results from applications dealing with heat and mass transfer, conduction-convection phenomena, and thermal treatments.

@ARTICLE{QuarteroniRozzaManzoni2011,
author = {Quarteroni, A. and Rozza, G. and Manzoni, A.},
title = {Certified reduced basis approximation for parametrized partial differential
equations and applications},
journal = {Journal of Mathematics in Industry},
year = {2011},
volume = {1},
pages = {1--49},
number = {1},
abstract = {Reduction strategies, such as model order reduction (MOR) or reduced
basis (RB) methods, in scientific computing may become crucial in
applications of increasing complexity. In this paper we review the
reduced basis method (built upon a high-fidelity truth'' finite element
approximation) for a rapid and reliable approximation of parametrized
partial differential equations, and comment on their potential impact
on applications of industrial interest. The essential ingredients
of RB methodology are: a Galerkin projection onto a low-dimensional
space of basis functions properly selected, an affine parametric
dependence enabling to perform a competitive Offline-Online splitting
in the computational procedure, and a rigorous a posteriori error
estimation used for both the basis selection and the certification
of the solution. The combination of these three factors yields substantial
computational savings which are at the basis of an efficient model
order reduction, ideally suited for real-time simulation and many-query
contexts (e.g. optimization, control or parameter identification).
After a brief excursus on the methodology, we focus on linear elliptic
and parabolic problems, discussing some extensions to more general
classes of problems and several perspectives of the ongoing research.
We present some results from applications dealing with heat and mass
transfer, conduction-convection phenomena, and thermal treatments.},
doi = {10.1186/2190-5983-1-3},
preprint = {http://mathicse.epfl.ch/files/content/sites/mathicse/files/Mathicse%20reports%202011/02.2011%20AQ-GR-AM.pdf}
}

5. G. Rozza, "Reduced basis approximation and error bounds for potential flows in parametrized geometries", Communications in Computational Physics, 9(1), pp. 1–48, 2011.
In this paper we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for potential flows in affinely parametrized geometries. We review the essential ingredients: i) a Galerkin projection onto a low-dimensional space associated with a smooth parametric manifold'' in order to get a dimension reduction; ii) an efficient and effective greedy sampling method for identification of optimal and numerically stable approximations to have a rapid convergence; iii) an a posteriori error estimation procedure: rigorous and sharp bounds for the linearfunctional outputs of interest and over the potential solution or related quantities of interest like velocity and/or pressure; iv) an Offline-Online computational decomposition strategies to achieve a minimum marginal computational cost for high performance in the real-time and many-query (e.g., design and optimization) contexts. We present three illustrative results for inviscid potential flows in parametrized geometries representing a Venturi channel, a circular bend and an added mass problem.

@ARTICLE{Rozza2011,
author = {Rozza, G.},
title = {Reduced basis approximation and error bounds for potential flows
in parametrized geometries},
journal = {Communications in Computational Physics},
year = {2011},
volume = {9},
pages = {1--48},
number = {1},
abstract = {In this paper we consider (hierarchical, Lagrange) reduced basis approximation
and a posteriori error estimation for potential flows in affinely
parametrized geometries. We review the essential ingredients: i)
a Galerkin projection onto a low-dimensional space associated with
a smooth parametric manifold'' in order to get a dimension reduction;
ii) an efficient and effective greedy sampling method for identification
of optimal and numerically stable approximations to have a rapid
convergence; iii) an a posteriori error estimation procedure: rigorous
and sharp bounds for the linearfunctional outputs of interest and
over the potential solution or related quantities of interest like
velocity and/or pressure; iv) an Offline-Online computational decomposition
strategies to achieve a minimum marginal computational cost for high
performance in the real-time and many-query (e.g., design and optimization)
contexts. We present three illustrative results for inviscid potential
flows in parametrized geometries representing a Venturi channel,
a circular bend and an added mass problem.},
doi = {l0.4208/cicp.l003l0.2607l0a},
preprint = {https://infoscience.epfl.ch/record/150152/files/2)%2011.2010.pdf}
}

6. G. Rozza, T. Lassila, and A. Manzoni, "Reduced basis approximation for shape optimization in thermal flows with a parametrized polynomial geometric map", Lecture Notes in Computational Science and Engineering, 76, pp. 307–315, 2011.
Reduced basis approximations for geometrically parametrized advection-diffusion equations are investigated. The parametric domains are assumed to be images of a reference domain through a piecewise polynomial map; this may lead to nonaffinely parametrized diffusion tensors that are treated with an empirical interpolation method. An a posteriori error bound including a correction term due to this approximation is given. Results concerning the applied methodology and the rigor of the corrected error estimator are shown for a shape optimization problem in a thermal flow.

@ARTICLE{RozzaLassilaManzoni2011,
author = {Rozza, G. and Lassila, T. and Manzoni, A.},
title = {Reduced basis approximation for shape optimization in thermal flows
with a parametrized polynomial geometric map},
journal = {Lecture Notes in Computational Science and Engineering},
year = {2011},
volume = {76},
pages = {307--315},
abstract = {Reduced basis approximations for geometrically parametrized advection-diffusion
equations are investigated. The parametric domains are assumed to
be images of a reference domain through a piecewise polynomial map;
this may lead to nonaffinely parametrized diffusion tensors that
are treated with an empirical interpolation method. An a posteriori
error bound including a correction term due to this approximation
is given. Results concerning the applied methodology and the rigor
of the corrected error estimator are shown for a shape optimization
problem in a thermal flow.},
doi = {10.1007/978-3-642-15337-2_28},
preprint = {https://infoscience.epfl.ch/record/146655/files/rozza_lassila_manzoni_reviewed.pdf}
}

### 2010

1. T. Lassila and G. Rozza, "Parametric free-form shape design with PDE models and reduced basis method", Computer Methods in Applied Mechanics and Engineering, 199(23-24), pp. 1583–1592, 2010.
We present a coupling of the reduced basis methods and free-form deformations for shape optimization and design of systems modelled by elliptic PDEs. The free-form deformations give a parameterization of the shape that is independent of the mesh, the initial geometry, and the underlying PDE model. The resulting parametric PDEs are solved by reduced basis methods. An important role in our implementation is played by the recently proposed empirical interpolation method, which allows approximating the non-affinely parameterized deformations with affinely parameterized ones. These ingredients together give rise to an efficient online computational procedure for a repeated evaluation design environment like the one for shape optimization. The proposed approach is demonstrated on an airfoil inverse design problem.

@ARTICLE{LassilaRozza2010a,
author = {Lassila, T. and Rozza, G.},
title = {Parametric free-form shape design with {PDE} models and reduced basis
method},
journal = {Computer Methods in Applied Mechanics and Engineering},
year = {2010},
volume = {199},
pages = {1583--1592},
number = {23-24},
abstract = {We present a coupling of the reduced basis methods and free-form deformations
for shape optimization and design of systems modelled by elliptic
PDEs. The free-form deformations give a parameterization of the shape
that is independent of the mesh, the initial geometry, and the underlying
PDE model. The resulting parametric PDEs are solved by reduced basis
methods. An important role in our implementation is played by the
recently proposed empirical interpolation method, which allows approximating
the non-affinely parameterized deformations with affinely parameterized
ones. These ingredients together give rise to an efficient online
computational procedure for a repeated evaluation design environment
like the one for shape optimization. The proposed approach is demonstrated
on an airfoil inverse design problem.},
doi = {10.1016/j.cma.2010.01.007},
preprint = {https://infoscience.epfl.ch/record/143436/files/ffdparam_revised_100108.pdf}
}

2. T. M. Lassila and G. Rozza, "Reduced formulation of a steady fluid-structure interaction problem with parametric coupling", in Proceedings of the 10th Finnish Mechanics Days Conference, 2010.
[BibTeX] [Abstract]
We propose a two-fold approach to model reduction of fluid-structure interaction. The state equations for the fluid are solved with reduced basis methods. These are model reduction methods for parametric partial differential equations using well-chosen snapshot solutions in order to build a set of global basis functions. The other reduction is in terms of the geometric complexity of the moving fluidstructure interface. We use free-form deformations to parameterize the perturbation of the flow channel at rest configuration. As a computational example we consider a steady fluid-structure interaction problem: an incompressible Stokes flow in a channel that has a flexible wall.

@INPROCEEDINGS{LassilaRozza2010,
author = {Lassila, Toni Mikael and Rozza, Gianluigi},
title = {Reduced formulation of a steady fluid-structure interaction problem
with parametric coupling},
booktitle = {Proceedings of the 10th {F}innish {M}echanics {D}ays {C}onference},
year = {2010},
editor = {Makinen, R. A. E. and Valpe, K. and Neittaanmaki, P. and Tuovinen,
T.},
abstract = {We propose a two-fold approach to model reduction of fluid-structure
interaction. The state equations for the fluid are solved with reduced
basis methods. These are model reduction methods for parametric partial
differential equations using well-chosen snapshot solutions in order
to build a set of global basis functions. The other reduction is
in terms of the geometric complexity of the moving fluidstructure
interface. We use free-form deformations to parameterize the perturbation
of the flow channel at rest configuration. As a computational example
we consider a steady fluid-structure interaction problem: an incompressible
Stokes flow in a channel that has a flexible wall.}
}

3. N. C. Nguyen, G. Rozza, D. B. P. Huynh, and A. T. Patera, "Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Parabolic PDEs: Application to Real-Time Bayesian Parameter Estimation", Large-Scale Inverse Problems and Quantification of Uncertainty, pp. 151–177, 2010.
@ARTICLE{NguyenRozzaHuynhPatera2010,
title = {Reduced Basis Approximation and a Posteriori Error Estimation for
Parametrized Parabolic {PDEs}: Application to Real-Time {B}ayesian
Parameter Estimation},
year = {2010},
author = {Nguyen, N.C. and Rozza, G. and Huynh, D.B.P. and Patera, A.T.},
pages = {151--177},
doi = {10.1002/9780470685853.ch8},
journal = {Large-Scale Inverse Problems and Quantification of Uncertainty},
preprint = {https://infoscience.epfl.ch/record/125956/files/atpWileyAug2008preprint.pdf}
}

4. G. Rozza and A. Manzoni, "Model Order Reduction by geometrical parametrization for shape optimization in computational fluid dynamics", in Proceedings of the ECCOMAS CFD 2010, V European Conference on Computational Fluid Dynamics, 2010.
[BibTeX] [Abstract]
Shape Optimization problems governed by partial differential equations result from many applications in computational fluid dynamics; they involve the repetitive evaluation of outputs expressed as functionals of the field variables and usually imply big computational efforts. For this reason looking for computational efficiency in numerical methods and algorithms is mandatory. The interplay between scientific computing and new reduction strategies is crucial in applications of great complexity. In order to achieve an efficient model order reduction, reduced basis methods built upon a high-fidelity truth'' finite element approximation – and combined with suitable geometrical parametrization techniques for efficient shape description – can be introduced, thus decreasing both the computational effort and the geometrical complexity. Starting from an excursus on classical approaches – such as local boundary variation and shape boundary parametrization – we focus on more efficient parametrization techniques which are well suited for a combination with a reduced basis approach, such as the one based on affine mapping (even automatic), nonaffine mapping (coupled with a suitable empirical interpolation technique for better numerical performances) and free-form deformations. We thus describe (and compare) the principal features of these parametrization techniques by showing some applications dealing with shape optimization of parametrized configurations in viscous flows,and discussing computational advantages and efficiency obtained by geometrical and computational model order reduction.

@INPROCEEDINGS{RozzaManzoni2010,
author = {Rozza, Gianluigi and Manzoni, Andrea},
title = {Model {O}rder {R}eduction by geometrical parametrization for shape
optimization in computational fluid dynamics},
booktitle = {Proceedings of the {ECCOMAS} {CFD} 2010, {V} {E}uropean {C}onference
on {C}omputational {F}luid {D}ynamics},
year = {2010},
editor = {Pereira, J. C. F. and Sequeira, Ad\'elia},
abstract = {Shape Optimization problems governed by partial differential equations
result from many applications in computational fluid dynamics; they
involve the repetitive evaluation of outputs expressed as functionals
of the field variables and usually imply big computational efforts.
For this reason looking for computational efficiency in numerical
methods and algorithms is mandatory. The interplay between scientific
computing and new reduction strategies is crucial in applications
of great complexity. In order to achieve an efficient model order
reduction, reduced basis methods built upon a high-fidelity truth''
finite element approximation -- and combined with suitable geometrical
parametrization techniques for efficient shape description -- can
be introduced, thus decreasing both the computational effort and
the geometrical complexity. Starting from an excursus on classical
approaches -- such as local boundary variation and shape boundary
parametrization -- we focus on more efficient parametrization techniques
which are well suited for a combination with a reduced basis approach,
such as the one based on affine mapping (even automatic), nonaffine
mapping (coupled with a suitable empirical interpolation technique
for better numerical performances) and free-form deformations. We
thus describe (and compare) the principal features of these parametrization
techniques by showing some applications dealing with shape optimization
of parametrized configurations in viscous flows,and discussing computational
advantages and efficiency obtained by geometrical and computational
model order reduction. }
}

5. Z. C. Xuan, T. Lassila, G. Rozza, and A. Quarteroni, "On computing upper and lower bounds on the outputs of linear elasticity problems approximated by the smoothed finite element method", International Journal for Numerical Methods in Engineering, 83(2), pp. 174–195, 2010.
Verification of the computation of local quantities of interest, e.g. the displacements at a point, the stresses in a local area and the stress intensity factors at crack tips, plays an important role in improving the structural design for safety. In this paper, the smoothed finite element method (SFEM) is used for finding upper and lower bounds on the local quantities of interest that are outputs of the displacement field for linear elasticity problems, based on bounds on strain energy in both the primal and dual problems. One important feature of SFEM is that it bounds the strain energy of the structure from above without needing the solutions of different subproblems that are based on elements or patches but only requires the direct finite element computation. Upper and lower bounds on two linear outputs and one quadratic output related with elasticity-the local reaction, the local displacement and the J-integral-are computed by the proposed method in two different examples. Some issues with SFEM that remain to be resolved are also discussed.

@ARTICLE{XuanLassilaRozzaQuarteroni2010,
author = {Xuan, Z.C. and Lassila, T. and Rozza, G. and Quarteroni, A.},
title = {On computing upper and lower bounds on the outputs of linear elasticity
problems approximated by the smoothed finite element method},
journal = {International Journal for Numerical Methods in Engineering},
year = {2010},
volume = {83},
pages = {174--195},
number = {2},
abstract = {Verification of the computation of local quantities of interest, e.g.
the displacements at a point, the stresses in a local area and the
stress intensity factors at crack tips, plays an important role in
improving the structural design for safety. In this paper, the smoothed
finite element method (SFEM) is used for finding upper and lower
bounds on the local quantities of interest that are outputs of the
displacement field for linear elasticity problems, based on bounds
on strain energy in both the primal and dual problems. One important
feature of SFEM is that it bounds the strain energy of the structure
from above without needing the solutions of different subproblems
that are based on elements or patches but only requires the direct
finite element computation. Upper and lower bounds on two linear
outputs and one quadratic output related with elasticity-the local
reaction, the local displacement and the J-integral-are computed
by the proposed method in two different examples. Some issues with
SFEM that remain to be resolved are also discussed.},
doi = {10.1002/nme.2825},
preprint = {https://infoscience.epfl.ch/record/140717/files/MOX_report_Xuan_Lassila_Rozza_Quarteroni.pdf}
}

### 2009

1. S. Deparis and G. Rozza, "Reduced basis method for multi-parameter-dependent steady Navier-Stokes equations: Applications to natural convection in a cavity", Journal of Computational Physics, 228(12), pp. 4359–4378, 2009.
This work focuses on the approximation of parametric steady Navier-Stokes equations by the reduced basis method. For a particular instance of the parameters under consideration, we are able to solve the underlying partial differential equations, compute an output, and give sharp error bounds. The computations are split into an offline part, where the values of the parameters are not yet identified, but only given within a range of interest, and an online part, where the problem is solved for an instance of the parameters. The offline part is expensive and is used to build a reduced basis and prepare all the ingredients - mainly matrix-vector and scalar products, but also eigenvalue computations - necessary for the online part, which is fast. We provide a model problem - describing natural convection phenomena in a laterally heated cavity - characterized by three parameters: Grashof and Prandtl numbers and the aspect ratio of the cavity. We show the feasibility and efficiency of the a posteriori error estimation by the natural norm approach considering several test cases by varying two different parameters. The gain in terms of CPU time with respect to a parallel finite element approximation is of three magnitude orders with an acceptable - indeed less than 0.1\% - error on the selected outputs.

@ARTICLE{DeparisRozza2009,
author = {Deparis, S. and Rozza, G.},
title = {Reduced basis method for multi-parameter-dependent steady {N}avier-{S}tokes
equations: Applications to natural convection in a cavity},
journal = {Journal of Computational Physics},
year = {2009},
volume = {228},
pages = {4359--4378},
number = {12},
abstract = {This work focuses on the approximation of parametric steady Navier-Stokes
equations by the reduced basis method. For a particular instance
of the parameters under consideration, we are able to solve the underlying
partial differential equations, compute an output, and give sharp
error bounds. The computations are split into an offline part, where
the values of the parameters are not yet identified, but only given
within a range of interest, and an online part, where the problem
is solved for an instance of the parameters. The offline part is
expensive and is used to build a reduced basis and prepare all the
ingredients - mainly matrix-vector and scalar products, but also
eigenvalue computations - necessary for the online part, which is
fast. We provide a model problem - describing natural convection
phenomena in a laterally heated cavity - characterized by three parameters:
Grashof and Prandtl numbers and the aspect ratio of the cavity. We
show the feasibility and efficiency of the a posteriori error estimation
by the natural norm approach considering several test cases by varying
two different parameters. The gain in terms of CPU time with respect
to a parallel finite element approximation is of three magnitude
orders with an acceptable - indeed less than 0.1\% - error on the
selected outputs.},
doi = {10.1016/j.jcp.2009.03.008},
preprint = {https://infoscience.epfl.ch/record/128722/files/Deparis_Rozza_JCP.pdf}
}

2. N. -C. Nguyen, G. Rozza, and A. T. Patera, "Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers' equation", Calcolo, 46(3), pp. 157–185, 2009.
In this paper we present rigorous a posteriori L2 error bounds for reduced basis approximations of the unsteady viscous Burgers' equation in one space dimension. The a posteriori error estimator, derived from standard analysis of the error-residual equation, comprises two key ingredients-both of which admit efficient Offline-Online treatment: the first is a sum over timesteps of the square of the dual norm of the residual; the second is an accurate upper bound (computed by the Successive Constraint Method) for the exponential-in-time stability factor. These error bounds serve both Offline for construction of the reduced basis space by a new POD-Greedy procedure and Online for verification of fidelity. The a posteriori error bounds are practicable for final times (measured in convective units) T=O(1) and Reynolds numbers nu-1 >> 1; we present numerical results for a (stationary) steepening front for T = 2 and 1$łe$nu-1$łe$200.

@ARTICLE{NguyenRozzaPatera2009,
author = {Nguyen, N.-C. and Rozza, G. and Patera, A.T.},
title = {Reduced basis approximation and a posteriori error estimation for
the time-dependent viscous {B}urgers' equation},
journal = {Calcolo},
year = {2009},
volume = {46},
pages = {157--185},
number = {3},
abstract = {In this paper we present rigorous a posteriori L2 error bounds for
reduced basis approximations of the unsteady viscous Burgers' equation
in one space dimension. The a posteriori error estimator, derived
from standard analysis of the error-residual equation, comprises
two key ingredients-both of which admit efficient Offline-Online
treatment: the first is a sum over timesteps of the square of the
dual norm of the residual; the second is an accurate upper bound
(computed by the Successive Constraint Method) for the exponential-in-time
stability factor. These error bounds serve both Offline for construction
of the reduced basis space by a new POD-Greedy procedure and Online
for verification of fidelity. The a posteriori error bounds are practicable
for final times (measured in convective units) T=O(1) and Reynolds
numbers nu-1 >> 1; we present numerical results for a (stationary)
steepening front for T = 2 and 1$\le$nu-1$\le$200.},
doi = {10.1007/s10092-009-0005-x},
preprint = {http://augustine.mit.edu/methodology/papers/atpCalcoloApr2009preprint.pdf}
}

3. G. Rozza, "Reduced basis methods for Stokes equations in domains with non-affine parameter dependence", Computing and Visualization in Science, 12(1), pp. 23–35, 2009.
[BibTeX] [Abstract] [View on publisher website]
In this paper we deal with reduced basis techniques applied to Stokes equations. We consider domains with different shape, parametrized by affine and non-affine maps with respect to a reference domain. The proposed method is ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. An empirical, stable and inexpensive interpolation procedure has permitted to replace non-affine coefficient functions with an expansion which leads to a computational decomposition between the off-line (parameter independent) stage for reduced basis generation and the on-line (parameter dependent) approximation stage based on Galerkin projection, used to find a new solution for a new set of parameters by a combination of previously computed stored solutions. As in the affine case this computational decomposition leads us to preserve reduced basis properties: rapid and accurate convergence and computational economies. The applications and results are based on parametrized geometries describing domains with curved walls, for example a stenosed channel and a bypass configuration. This method is well suited to treat also problems in fixed domain with non-affine parameters dependence expressing varying physical coefficients.

@ARTICLE{Rozza2009,
author = {Rozza, G.},
title = {Reduced basis methods for {S}tokes equations in domains with non-affine
parameter dependence},
journal = {Computing and Visualization in Science},
year = {2009},
volume = {12},
pages = {23--35},
number = {1},
abstract = {In this paper we deal with reduced basis techniques applied to Stokes
equations. We consider domains with different shape, parametrized
by affine and non-affine maps with respect to a reference domain.
The proposed method is ideally suited for the repeated and rapid
evaluations required in the context of parameter estimation, design,
optimization, and real-time control. An empirical, stable and inexpensive
interpolation procedure has permitted to replace non-affine coefficient
functions with an expansion which leads to a computational decomposition
between the off-line (parameter independent) stage for reduced basis
generation and the on-line (parameter dependent) approximation stage
based on Galerkin projection, used to find a new solution for a new
set of parameters by a combination of previously computed stored
solutions. As in the affine case this computational decomposition
leads us to preserve reduced basis properties: rapid and accurate
convergence and computational economies. The applications and results
are based on parametrized geometries describing domains with curved
walls, for example a stenosed channel and a bypass configuration.
This method is well suited to treat also problems in fixed domain
with non-affine parameters dependence expressing varying physical
coefficients.},
doi = {10.1007/s00791-006-0044-7}
}

4. G. Rozza, "An introduction to reduced basis method for parametrized PDEs", in Applied and Industrial Mathematics in Italy, 2009, pp. 508–519.
We provide an introduction on reduced basis (RB) method for the solution of parameterized partial differential equations (PDEs). We introduce all the main ingredients to describe the methodology and the algorithms used to build the approximation spaces and the error bounds. We consider a model problem describing a steady potential flow around parametrized bodies and we provide some illustrative results.

@INPROCEEDINGS{Rozza2009a,
author = {Rozza, Gianluigi},
title = {An introduction to reduced basis method for parametrized {PDE}s},
booktitle = {Applied and {I}ndustrial {M}athematics in {I}taly},
year = {2009},
editor = {De Bernardis, Enrico and Spiegler, Renato and Valente, Vanda},
volume = {III},
series = {Advances in Mathematics for Applied Sciences},
pages = {508--519},
abstract = {We provide an introduction on reduced basis (RB) method for the solution
of parameterized partial differential equations (PDEs). We introduce
all the main ingredients to describe the methodology and the algorithms
used to build the approximation spaces and the error bounds. We consider
a model problem describing a steady potential flow around parametrized
bodies and we provide some illustrative results.},
preprint = {https://infoscience.epfl.ch/record/138787/files/Rozza_AIMI_III.pdf}
}

5. G. Rozza, D. B. P. Huynh, N. C. Nguyen, and A. T. Patera, "Real-time reliable simulation of heat transfer phenomena", in Proceedings of the ASME Summer Heat Transfer Conference 2009, HT2009, 2009, pp. 851–860.
In this paper we discuss the application of the certified reduced basis method and the associated software package rbMIT to worked problems'' in steady and unsteady conduction. Each worked problem is characterized by an input parameter vector - material properties, boundary conditions and sources, and geometry - and desired outputs - selected fluxes and temperatures. The methodology and associated rbMIT software, as well as the educational worked problem framework, consists of two distinct stages: an Offline (or Instructor'') stage in which a new heat transfer worked problem is first created; and an Online (or Lecturer''/ Student'') stage in which the worked problem is subsequently invoked in (say) various inclass, project, or homework settings. In the very inexpensive Online stage, given an input parameter value, the software returns both (i) an accurate reduced basis output prediction, and (ii) a rigorous bound for the error in the reduced basis prediction relative to an underlying expensive high-fidelity finite element discretization; as required in the educational context, the response is both rapid and reliable. We present illustrative results for two worked problems: a steady thermal fin, and unsteady thermal analysis of a delamination crack.

@INPROCEEDINGS{RozzaHuynhNguyenPatera2009,
author = {Rozza, G. and Huynh, D.B.P. and Nguyen, N.C. and Patera, A.T.},
title = {Real-time reliable simulation of heat transfer phenomena},
year = {2009},
volume = {3},
pages = {851--860},
abstract = {In this paper we discuss the application of the certified reduced
basis method and the associated software package rbMIT to worked
is characterized by an input parameter vector - material properties,
boundary conditions and sources, and geometry - and desired outputs
- selected fluxes and temperatures. The methodology and associated
rbMIT software, as well as the educational worked problem framework,
consists of two distinct stages: an Offline (or Instructor'') stage
in which a new heat transfer worked problem is first created; and
an Online (or Lecturer''/ Student'') stage in which the worked problem
is subsequently invoked in (say) various inclass, project, or homework
settings. In the very inexpensive Online stage, given an input parameter
value, the software returns both (i) an accurate reduced basis output
prediction, and (ii) a rigorous bound for the error in the reduced
basis prediction relative to an underlying expensive high-fidelity
finite element discretization; as required in the educational context,
the response is both rapid and reliable. We present illustrative
thermal analysis of a delamination crack.},
doi = {10.1115/HT2009-88212},
booktitle = {Proceedings of the ASME Summer Heat Transfer Conference 2009, HT2009},
preprint = {https://infoscience.epfl.ch/record/137298/files/ASME-HT-2009-88212.pdf}
}

6. G. Rozza, C. N. Nguyen, A. T. Patera, and S. Deparis, "Reduced basis methods and a posteriori error estimators for heat transfer problems", in Proceedings of the ASME Summer Heat Transfer Conference 2009, HT2009, 2009, pp. 753–762.
This paper focuses on the parametric study of steady and unsteady forced and natural convection problems by the certified reduced basis method. These problems are characterized by an input-output relationship in which given an input parameter vector - material properties, boundary conditions and sources, and geometry - we would like to compute certain outputs of engineering interest - heat fluxes and average temperatures. The certified reduced basis method provides both (i) a very inexpensive yet accurate output prediction, and (ii) a rigorous bound for the error in the reduced basis prediction relative to an underlying expensive high-fidelity finite element discretization. The feasibility and efficiency of the method is demonstrated for three natural convection model problems: a scalar steady forced convection problem in a rectangular channel is characterized by two parameters - Péclet number and the aspect ratio of the channel - and an output - the average temperature over the domain; a steady natural convection problem in a laterally heated cavity is characterized by three parameters -Grashof and Prandtl numbers, and the aspect ratio of the cavity - and an output - the inverse of the Nusselt number; and an unsteady natural convection problem in a laterally heated cavity is characterized by two parameters -Grashof and Prandtl numbers - and a time-dependent output - the average of the horizontal velocity over a specified area of the cavity.

@INPROCEEDINGS{RozzaNguyenPateraDeparis2009,
author = {Rozza, G. and Nguyen, C.N. and Patera, A.T. and Deparis, S.},
title = {Reduced basis methods and a posteriori error estimators for heat
transfer problems},
year = {2009},
volume = {2},
pages = {753--762},
abstract = {This paper focuses on the parametric study of steady and unsteady
forced and natural convection problems by the certified reduced basis
method. These problems are characterized by an input-output relationship
in which given an input parameter vector - material properties, boundary
conditions and sources, and geometry - we would like to compute certain
outputs of engineering interest - heat fluxes and average temperatures.
The certified reduced basis method provides both (i) a very inexpensive
yet accurate output prediction, and (ii) a rigorous bound for the
error in the reduced basis prediction relative to an underlying expensive
high-fidelity finite element discretization. The feasibility and
efficiency of the method is demonstrated for three natural convection
model problems: a scalar steady forced convection problem in a rectangular
channel is characterized by two parameters - P\'eclet number and the
aspect ratio of the channel - and an output - the average temperature
over the domain; a steady natural convection problem in a laterally
heated cavity is characterized by three parameters -Grashof and Prandtl
numbers, and the aspect ratio of the cavity - and an output - the
inverse of the Nusselt number; and an unsteady natural convection
problem in a laterally heated cavity is characterized by two parameters
-Grashof and Prandtl numbers - and a time-dependent output - the
average of the horizontal velocity over a specified area of the cavity.},
doi = {10.1115/HT2009-88211},
booktitle = {Proceedings of the ASME Summer Heat Transfer Conference 2009, HT2009},
preprint = {https://infoscience.epfl.ch/record/138604/files/ASME-HT-2009-88211_R.pdf}
}

7. G. Rozza and A. Quarteroni, "Reduced basis approximation for parametrized partial differential equations", in Numerical Models for Differential Equations, E. Zuazua, C. Le Bris, A. T. Patera, A. Quarteroni, and T. Hou (eds.), Springer, vol. 2, pp. 556–587, 2009.
[BibTeX] [Abstract]
In this chapter we describe the basic ideas of reduced basis (RB) approximation methods for rapid and reliable evaluation of input-output relationships in which the output is expressed as a functional of a field variable that is the solution of an input-parametrized partial differential equation (PDE). We shall focus on linear output functionals and affinely parametrized linear elliptic coercive PDEs.

@INCOLLECTION{RozzaQuarteroni2009,
author = {Rozza, Gianluigi and Quarteroni, Alfio},
title = {Reduced basis approximation for parametrized partial differential
equations},
booktitle = {Numerical {M}odels for {D}ifferential {E}quations},
publisher = {Springer},
year = {2009},
editor = {Zuazua, E. and Le Bris, C. and Patera, A.T and Quarteroni, A. and
Hou, T.},
volume = {2},
series = {MS\&A, Modelling, Simulation and Application},
pages = {556--587},
abstract = {In this chapter we describe the basic ideas of reduced basis (RB)
approximation methods for rapid and reliable evaluation of input-output
relationships in which the output is expressed as a functional of
a field variable that is the solution of an input-parametrized partial
differential equation (PDE). We shall focus on linear output functionals
and affinely parametrized linear elliptic coercive PDEs.}
}

### 2008

1. B. Haasdonk, M. Ohlberger, and G. Rozza, "A reduced basis method for evolution schemes with parameter-dependent explicit operators", Electronic Transactions on Numerical Analysis, 32, pp. 145–161, 2008.
During the last decades, reduced basis (RB) methods have been developed to a wide methodology for model reduction of problems that are governed by parametrized partial differential equations (PDEs). In particular equations of elliptic and parabolic type for linear, low degree polynomial or monotonic nonlinearities have been treated successfully by RB methods using finite element schemes. Due to the characteristic offline-online decomposition, the reduced models often become suitable for a multi-query or real-time setting, where simulation results, such as field-variables or output estimates, can be approximated reliably and rapidly for varying parameters. In the current study, we address a certain class of time-dependent evolution schemes with explicit discretization operators that are arbitrarily parameter dependent. We extend the RB methodology to these cases by applying the empirical interpolation method to localized discretization operators. The main technical ingredients are: (i) generation of a collateral reduced basis modelling the effects of the discretization operator under parameter variations in the offline-phase and (ii) an online simulation scheme based on a numerical subgrid and localized evaluations of the evolution operator. We formulate an a-posteriori error estimator for quantification of the resulting reduced simulation error. Numerical experiments on a parametrized convection problem, discretized with a finite volume scheme, demonstrate the applicability of the model reduction technique. We obtain a parametrized reduced model, which enables parameter variation with fast simulation response. We quantify the computational gain with respect to the non-reduced model and investigate the error convergence.

@ARTICLE{HaasdonkOhlbergerRozza2008,
author = {Haasdonk, B. and Ohlberger, M. and Rozza, G.},
title = {A reduced basis method for evolution schemes with parameter-dependent
explicit operators},
journal = {Electronic Transactions on Numerical Analysis},
year = {2008},
volume = {32},
pages = {145--161},
abstract = {During the last decades, reduced basis (RB) methods have been developed
to a wide methodology for model reduction of problems that are governed
by parametrized partial differential equations (PDEs). In particular
equations of elliptic and parabolic type for linear, low degree polynomial
or monotonic nonlinearities have been treated successfully by RB
methods using finite element schemes. Due to the characteristic offline-online
decomposition, the reduced models often become suitable for a multi-query
or real-time setting, where simulation results, such as field-variables
or output estimates, can be approximated reliably and rapidly for
varying parameters. In the current study, we address a certain class
of time-dependent evolution schemes with explicit discretization
operators that are arbitrarily parameter dependent. We extend the
RB methodology to these cases by applying the empirical interpolation
method to localized discretization operators. The main technical
ingredients are: (i) generation of a collateral reduced basis modelling
the effects of the discretization operator under parameter variations
in the offline-phase and (ii) an online simulation scheme based on
a numerical subgrid and localized evaluations of the evolution operator.
We formulate an a-posteriori error estimator for quantification of
the resulting reduced simulation error. Numerical experiments on
a parametrized convection problem, discretized with a finite volume
scheme, demonstrate the applicability of the model reduction technique.
We obtain a parametrized reduced model, which enables parameter variation
with fast simulation response. We quantify the computational gain
with respect to the non-reduced model and investigate the error convergence.},
preprint={https://infoscience.epfl.ch/record/124834/files/chemnitz_ETNA_HaOhRo_revised.pdf},
doi={http://emis.ams.org/journals/ETNA/vol.32.2008/pp145-161.dir/pp145-161.pdf}
}

2. R. Milani, A. Quarteroni, and G. Rozza, "Reduced basis method for linear elasticity problems with many parameters", Computer Methods in Applied Mechanics and Engineering, 197(51-52), pp. 4812–4829, 2008.
The reduced basis (RB) methods are proposed here for the solution of parametrized equations in linear elasticity problems. The fundamental idea underlying RB methods is to decouple the generation and projection stages (offline/online computational procedures) of the approximation process in order to solve parametrized equations in a rapid, inexpensive and reliable way. The method allows important computational savings with respect to the classical Galerkin-finite element method, ill suited to a repetitive environment like the parametrized contexts of optimization, many queries and sensitivity analysis. We consider different parametrization for the systems: either physical quantities - to model the materials and loads - and geometrical parameters - to model different geometrical configurations. Then we describe three different applications of the method in problems with isotropic and orthotropic materials working in plane stress and plane strain approximation and subject to harmonic loads.

@ARTICLE{MilaniQuarteroniRozza2008,
author = {Milani, R. and Quarteroni, A. and Rozza, G.},
title = {Reduced basis method for linear elasticity problems with many parameters},
journal = {Computer Methods in Applied Mechanics and Engineering},
year = {2008},
volume = {197},
pages = {4812--4829},
number = {51-52},
abstract = {The reduced basis (RB) methods are proposed here for the solution
of parametrized equations in linear elasticity problems. The fundamental
idea underlying RB methods is to decouple the generation and projection
stages (offline/online computational procedures) of the approximation
process in order to solve parametrized equations in a rapid, inexpensive
and reliable way. The method allows important computational savings
with respect to the classical Galerkin-finite element method, ill
suited to a repetitive environment like the parametrized contexts
of optimization, many queries and sensitivity analysis. We consider
different parametrization for the systems: either physical quantities
- to model the materials and loads - and geometrical parameters -
to model different geometrical configurations. Then we describe three
different applications of the method in problems with isotropic and
orthotropic materials working in plane stress and plane strain approximation
doi = {10.1016/j.cma.2008.07.002},
preprint = {https://infoscience.epfl.ch/record/125706/files/MQR.pdf}
}

3. G. Rozza, D. B. P. Huynh, and A. T. Patera, "Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: Application to transport and continuum mechanics", Archives of Computational Methods in Engineering, 15(3), pp. 229–275, 2008.
In this paper we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equations. The essential ingredients are (primal-dual) Galerkin projection onto a low-dimensional space associated with a smooth parametric manifold''-dimension reduction; efficient and effective greedy sampling methods for identification of optimal and numerically stable approximations-rapid convergence; a posteriori error estimation procedures-rigorous and sharp bounds for the linear-functional outputs of interest; and Offline-Online computational decomposition strategies-minimum marginal cost for high performance in the real-time/embedded (e.g., parameter-estimation, control) and many-query (e.g., design optimization, multi-model/scale) contexts. We present illustrative results for heat conduction and convection-diffusion, inviscid flow, and linear elasticity; outputs include transport rates, added mass, and stress intensity factors.

@ARTICLE{RozzaHuynhPatera2008,
author = {Rozza, G. and Huynh, D.B.P. and Patera, A.T.},
title = {Reduced basis approximation and a posteriori error estimation for
affinely parametrized elliptic coercive partial differential equations:
Application to transport and continuum mechanics},
journal = {Archives of Computational Methods in Engineering},
year = {2008},
volume = {15},
pages = {229--275},
number = {3},
abstract = {In this paper we consider (hierarchical, Lagrange) reduced basis approximation
and a posteriori error estimation for linear functional outputs of
affinely parametrized elliptic coercive partial differential equations.
The essential ingredients are (primal-dual) Galerkin projection onto
a low-dimensional space associated with a smooth parametric manifold''-dimension
reduction; efficient and effective greedy sampling methods for identification
of optimal and numerically stable approximations-rapid convergence;
a posteriori error estimation procedures-rigorous and sharp bounds
for the linear-functional outputs of interest; and Offline-Online
computational decomposition strategies-minimum marginal cost for
high performance in the real-time/embedded (e.g., parameter-estimation,
control) and many-query (e.g., design optimization, multi-model/scale)
contexts. We present illustrative results for heat conduction and
convection-diffusion, inviscid flow, and linear elasticity; outputs
include transport rates, added mass, and stress intensity factors.},
doi = {10.1007/s11831-008-9019-9},
preprint = {https://infoscience.epfl.ch/record/124831/files/ARCME.pdf}
}

### 2007

1. D. B. P. Huynh, G. Rozza, S. Sen, and A. T. Patera, "A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants", Comptes Rendus Mathematique, 345(8), pp. 473–478, 2007.
We present an approach to the construction of lower bounds for the coercivity and inf-sup stability constants required in a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations. The method, based on an Offline-Online strategy relevant in the reduced basis many-query and real-time context, reduces the Online calculation to a small Linear Program: the objective is a parametric expansion of the underlying Rayleigh quotient; the constraints reflect stability information at optimally selected parameter points. Numerical results are presented for coercive elasticity and non-coercive acoustics Helmholtz problems. To cite this article: D.B.P. Huynh et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).

@ARTICLE{HuynhRozzaSenPatera2007,
author = {Huynh, D.B.P. and Rozza, G. and Sen, S. and Patera, A.T.},
title = {A successive constraint linear optimization method for lower bounds
of parametric coercivity and inf-sup stability constants},
journal = {Comptes Rendus Mathematique},
year = {2007},
volume = {345},
pages = {473--478},
number = {8},
abstract = {We present an approach to the construction of lower bounds for the
coercivity and inf-sup stability constants required in a posteriori
error analysis of reduced basis approximations to affinely parametrized
partial differential equations. The method, based on an Offline-Online
strategy relevant in the reduced basis many-query and real-time context,
reduces the Online calculation to a small Linear Program: the objective
is a parametric expansion of the underlying Rayleigh quotient; the
constraints reflect stability information at optimally selected parameter
points. Numerical results are presented for coercive elasticity and
D.B.P. Huynh et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).},
doi = {10.1016/j.crma.2007.09.019},
preprint = {http://web.mit.edu/huynh/www/Papers/scm.pdf}
}

2. A. Quarteroni and G. Rozza, "Tecniche a Basi Ridotte per l'Ottimizzazione di Configurazioni di Innesto per Bypass Coronarici", in Un grande matematico dell'800: omaggio a Eugenio Beltrami, 2007, pp. 225–238.
[BibTeX]
@INPROCEEDINGS{QuarteroniRozza2007,
author = {Quarteroni, Alfio and Rozza, Gianluigi},
title = {Tecniche a {B}asi {R}idotte per l'{O}ttimizzazione di {C}onfigurazioni
di {I}nnesto per {B}ypass {C}oronarici},
booktitle = {Un grande matematico dell'800: omaggio a {E}ugenio {B}eltrami},
year = {2007},
e Lettere, Milano, Italy},
pages = {225--238}
}

3. A. Quarteroni, G. Rozza, and A. Quaini, "Reduced basis methods for optimal control of advection-diffusion problems", in Advances in Numerical Mathematics, 2007, pp. 193–216.
[BibTeX]
@INPROCEEDINGS{QuarteroniRozzaQuaini2007,
author = {Quarteroni, Alfio and Rozza, Gianluigi and Quaini, Annalisa},
title = {Reduced basis methods for optimal control of advection-diffusion
problems},
booktitle = {Advances in {N}umerical {M}athematics},
year = {2007},
editor = {Fitzgibbon, W. and Hoppe, R. and Periaux, J. and Pironneau, O. and
Vassilevski, Y.},
pages = {193--216}
}

4. A. Quarteroni and G. Rozza, "Numerical solution of parametrized Navier-Stokes equations by reduced basis methods", Numerical Methods for Partial Differential Equations, 23(4), pp. 923–948, 2007.
We apply the reduced basis method to solve Navier-Stokes equations in parametrized domains. Special attention is devoted to the treatment of the parametrized nonlinear transport term in the reduced basis framework, including the case of nonaffine parametric dependence that is treated by an empirical interpolation method. This method features (i) a rapid global convergence owing to the property of the Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N (optimally) selected points in the parameter space, and (ii) the offline/online computational procedures that decouple the generation and projection stages of the approximation process. This method is well suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. Our analysis focuses on: (i) the pressure treatment of incompressible Navier-Stokes problem; (ii) the fulfillment of an equivalent inf-sup condition to guarantee the stability of the reduced basis solutions. The applications that we consider involve parametrized geometries, like e.g. a channel with curved upper wall or an arterial bypass configuration.

@ARTICLE{QuarteroniRozza2007,
author = {Quarteroni, A. and Rozza, G.},
title = {Numerical solution of parametrized {N}avier-{S}tokes equations by
reduced basis methods},
journal = {Numerical Methods for Partial Differential Equations},
year = {2007},
volume = {23},
pages = {923--948},
number = {4},
abstract = {We apply the reduced basis method to solve Navier-Stokes equations
in parametrized domains. Special attention is devoted to the treatment
of the parametrized nonlinear transport term in the reduced basis
framework, including the case of nonaffine parametric dependence
that is treated by an empirical interpolation method. This method
features (i) a rapid global convergence owing to the property of
the Galerkin projection onto a space WN spanned by solutions of the
governing partial differential equation at N (optimally) selected
points in the parameter space, and (ii) the offline/online computational
procedures that decouple the generation and projection stages of
the approximation process. This method is well suited for the repeated
and rapid evaluations required in the context of parameter estimation,
design, optimization, and real-time control. Our analysis focuses
on: (i) the pressure treatment of incompressible Navier-Stokes problem;
(ii) the fulfillment of an equivalent inf-sup condition to guarantee
the stability of the reduced basis solutions. The applications that
we consider involve parametrized geometries, like e.g. a channel
with curved upper wall or an arterial bypass configuration.},
doi = {10.1002/num.20249},
}

5. G. Rozza and K. Veroy, "On the stability of the reduced basis method for Stokes equations in parametrized domains", Computer Methods in Applied Mechanics and Engineering, 196(7), pp. 1244–1260, 2007.
We present an application of reduced basis method for Stokes equations in domains with affine parametric dependence. The essential components of the method are (i) the rapid convergence of global reduced basis approximations - Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) the off-line/on-line computational procedures decoupling the generation and projection stages of the approximation process. The operation count for the on-line stage - in which, given a new parameter value, we calculate an output of interest - depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. Particular attention is given (i) to the pressure treatment of incompressible Stokes problem; (ii) to find an equivalent inf-sup condition that guarantees stability of reduced basis solutions by enriching the reduced basis velocity approximation space with the solutions of a supremizer problem; (iii) to provide algebraic stability of the problem by reducing the condition number of reduced basis matrices using an orthonormalization procedure applied to basis functions; (iv) to reduce computational costs in order to allow real-time solution of parametrized problem.

@ARTICLE{RozzaVeroy2007,
author = {Rozza, G. and Veroy, K.},
title = {On the stability of the reduced basis method for {S}tokes equations
in parametrized domains},
journal = {Computer Methods in Applied Mechanics and Engineering},
year = {2007},
volume = {196},
pages = {1244--1260},
number = {7},
abstract = {We present an application of reduced basis method for Stokes equations
in domains with affine parametric dependence. The essential components
of the method are (i) the rapid convergence of global reduced basis
approximations - Galerkin projection onto a space WN spanned by solutions
of the governing partial differential equation at N selected points
in parameter space; (ii) the off-line/on-line computational procedures
decoupling the generation and projection stages of the approximation
process. The operation count for the on-line stage - in which, given
a new parameter value, we calculate an output of interest - depends
only on N (typically very small) and the parametric complexity of
the problem; the method is thus ideally suited for the repeated and
rapid evaluations required in the context of parameter estimation,
design, optimization, and real-time control. Particular attention
is given (i) to the pressure treatment of incompressible Stokes problem;
(ii) to find an equivalent inf-sup condition that guarantees stability
of reduced basis solutions by enriching the reduced basis velocity
approximation space with the solutions of a supremizer problem; (iii)
to provide algebraic stability of the problem by reducing the condition
number of reduced basis matrices using an orthonormalization procedure
applied to basis functions; (iv) to reduce computational costs in
order to allow real-time solution of parametrized problem.},
doi = {10.1016/j.cma.2006.09.005},
preprint = {http://augustine.mit.edu/summerSchool08/rozza_CMAME2007.pdf}
}

### 2006

1. V. Agoshkov, A. Quarteroni, and G. Rozza, "Shape design in aorto-coronaric bypass anastomoses using perturbation theory", SIAM Journal on Numerical Analysis, 44(1), pp. 367–384, 2006.
In this paper we present a new approach in the study of aorto-coronaric bypass anastomoses configurations based on small perturbation theory. The theory of optimal control based on adjoint formulation is applied in order to optimize the shape of the zone of the incoming branch of the bypass (the toe) into the coronary (see Figure 2.1). The aim is to provide design indications in the perspective of future development for prosthetic bypasses.

@ARTICLE{AgoshkovQuarteroniRozza2006,
author = {Agoshkov, V. and Quarteroni, A. and Rozza, G.},
theory},
journal = {SIAM Journal on Numerical Analysis},
year = {2006},
volume = {44},
pages = {367--384},
number = {1},
abstract = {In this paper we present a new approach in the study of aorto-coronaric
bypass anastomoses configurations based on small perturbation theory.
The theory of optimal control based on adjoint formulation is applied
in order to optimize the shape of the zone of the incoming branch
of the bypass (the toe) into the coronary (see Figure 2.1). The aim
for prosthetic bypasses.},
doi = {10.1137/040613287},
preprint = {https://infoscience.epfl.ch/record/102999/files/AQR20031-new.pdf}
}

2. V. Agoshkov, A. Quarteroni, and G. Rozza, "A mathematical approach in the design of arterial bypass using unsteady Stokes equations", Journal of Scientific Computing, 28(2-3), pp. 139–165, 2006.
In this paper we present an approach for the study of Aorto-Coronaric bypass anastomoses configurations using unsteady Stokes equations. The theory of optimal control based on adjoint formulation is applied in order to optimize the shape of the zone of the incoming branch of the bypass (the toe) into the coronary according to several optimality criteria.

@ARTICLE{AgoshkovQuarteroniRozza2006a,
author = {Agoshkov, V. and Quarteroni, A. and Rozza, G.},
title = {A mathematical approach in the design of arterial bypass using unsteady
{S}tokes equations},
journal = {Journal of Scientific Computing},
year = {2006},
volume = {28},
pages = {139--165},
number = {2-3},
abstract = {In this paper we present an approach for the study of Aorto-Coronaric
bypass anastomoses configurations using unsteady Stokes equations.
The theory of optimal control based on adjoint formulation is applied
in order to optimize the shape of the zone of the incoming branch
of the bypass (the toe) into the coronary according to several optimality
criteria.},
doi = {10.1007/s10915-006-9077-9},
}

3. G. Fourestey, N. Parolini, C. Prud'homme, A. Quarteroni, and G. Rozza, "Matematica in volo con Solar Impulse", in Matematica e Cultura 2006, 2006, pp. 35–48.
[BibTeX]
@INPROCEEDINGS{FouresteyParoliniPrudhommeQuarteroniRozza2006,
author = {Fourestey, Gilles and Parolini, Nicola and Prud'homme, Christophe
and Quarteroni, Alfio and Rozza, Gianluigi},
title = {Matematica in volo con {S}olar {I}mpulse},
booktitle = {Matematica e {C}ultura 2006},
year = {2006},
pages = {35--48},
unit = {CMCS}
}

4. A. T. Patera and G. Rozza, "Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations. Version 1.0, Copyright MIT 2006-2007", to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering., 2006.
@UNPUBLISHED{PateraRozza2006,
author = {Patera, A. T. and Rozza, G.},
title = {Reduced Basis Approximation and A Posteriori Error Estimation for
Parametrized Partial Differential Equations. Version 1.0, Copyright
{MIT} 2006-2007},
year = {2006},
note = {to appear in (tentative rubric) MIT Pappalardo Graduate Monographs
in Mechanical Engineering.},
preprint = {http://augustine.mit.edu/methodology/methodology_bookPartI.htm}
}

5. A. Quarteroni, G. Rozza, L. Dedè, and A. Quaini, "Numerical approximation of a control problem for advection-diffusion processes", IFIP International Federation for Information Processing, 199, pp. 261–273, 2006.
Two different approaches are proposed to enhance the efficiency of the numerical resolution of optimal control problems governed by a linear advection-diffusion equation. In the framework of the Galerkin-Finite Element (FE) method, we adopt a novel a posteriori error estimate of the discretization error on the cost functional; this estimate is used in the course of a numerical adaptive strategy for the generation of efficient grids for the resolution of the optimal control problem. Moreover, we propose to solve the control problem by adopting a reduced basis (RB) technique, hence ensuring rapid, reliable and repeated evaluations of input-output relationship. Our numerical tests show that by this technique a substantial saving of computational costs can be achieved.

@ARTICLE{QuarteroniRozzaDedeQuaini2006,
author = {Quarteroni, A. and Rozza, G. and Ded\e, L. and Quaini, A.},
title = {Numerical approximation of a control problem for advection-diffusion
processes},
journal = {IFIP International Federation for Information Processing},
year = {2006},
volume = {199},
pages = {261--273},
abstract = {Two different approaches are proposed to enhance the efficiency of
the numerical resolution of optimal control problems governed by
a linear advection-diffusion equation. In the framework of the Galerkin-Finite
Element (FE) method, we adopt a novel a posteriori error estimate
of the discretization error on the cost functional; this estimate
is used in the course of a numerical adaptive strategy for the generation
of efficient grids for the resolution of the optimal control problem.
Moreover, we propose to solve the control problem by adopting a reduced
basis (RB) technique, hence ensuring rapid, reliable and repeated
evaluations of input-output relationship. Our numerical tests show
that by this technique a substantial saving of computational costs
can be achieved.},
doi = {10.1007/0-387-33006-2_24},
year = {2006},
}

6. G. Rozza, "Real-time reduced basis solutions for Navier-Stokes equations: optimization of parametrized bypass configurations", in ECCOMAS CFD 2006 Proceedings on CFD, 2006, pp. 1–16.
The reduced basis method on parametrized domains is applied to approximate blood flow through an arterial bypass. The aim is to provide (a) a sensitivity analysis for relevant geometrical quantities of interest in bypass configurations and (b) rapid and reliable prediction of integral functional outputs ( such as fluid mechanics indexes). The goal of this investigation is (i) to achieve design indications for arterial surgery in the perspective of future development for prosthetic bypasses, (ii) to develop numerical methods for optimization and design in biomechanics, and (iii) to provide an input-output relationship led by models with lower complexity and computational costs than the complete solution of fluid dynamics equations by a classical finite element method.

@INPROCEEDINGS{Rozza2006,
author = {Rozza, Gianluigi},
title = {Real-time reduced basis solutions for {N}avier-{S}tokes equations:
optimization of parametrized bypass configurations},
booktitle = {{ECCOMAS} {CFD} 2006 {P}roceedings on {CFD}},
editor = {Wesseling, P. and Onate, E. and Periaux, J.},
number = {676},
pages = {1--16},
abstract = {The reduced basis method on parametrized domains is applied to approximate
blood flow through an arterial bypass. The aim is to provide (a)
a sensitivity analysis for relevant geometrical quantities of interest
in bypass configurations and (b) rapid and reliable prediction of
integral functional outputs ( such as fluid mechanics indexes). The
arterial surgery in the perspective of future development for prosthetic
bypasses, (ii) to develop numerical methods for optimization and
led by models with lower complexity and computational costs than
the complete solution of fluid dynamics equations by a classical
finite element method.},
year = {2006},
preprint = {https://infoscience.epfl.ch/record/102986/files/294-371.pdf}
}

### 2005

1. G. Rozza, "Shape design by optimal flow control and reduced basis techniques: applications to bypass configurations in haemodynamics", PhD Thesis, École Polytechnique Fédérale de Lausanne, N. 3400, 2005.
[BibTeX]
@PHDTHESIS{Rozza2005a,
author = {Rozza, Gianluigi},
title = {Shape design by optimal flow control and reduced basis techniques:
applications to bypass configurations in haemodynamics},
school = {\'Ecole Polytechnique F\'ed\'erale de Lausanne, N. 3400},
year = {2005}
}

2. G. Rozza, "Reduced-basis methods for elliptic equations in sub-domains with a posteriori error bounds and adaptivity", Applied Numerical Mathematics, 55(4), pp. 403–424, 2005.
We present an application in multi-parametrized sub-domains based on a technique for the rapid and reliable prediction of linear-functional outputs of elliptic coercive partial differential equations with affine parameter dependence (reduced-basis methods). The main components are (i) rapidly convergent global reduced-basis approximations - Galerkin projection onto a space WN spanned by solutions of the governing equation at N selected points in parameter space (chosen by an adaptive procedure to minimize the estimated error and the effectivity; (ii) a posteriori error estimation - relaxations of the error-residual equation that provide inexpensive bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures - methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage - in which, given a new parameter value, we calculate the output of interest and associated error bound - depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. The application is based on a heat transfer problem in a parametrized geometry in view of haemodynamics applications and biomechanical devices optimization, such as the bypass configuration problem.

@ARTICLE{Rozza2005b,
author = {Rozza, G.},
title = {Reduced-basis methods for elliptic equations in sub-domains with
a posteriori error bounds and adaptivity},
journal = {Applied Numerical Mathematics},
year = {2005},
volume = {55},
pages = {403--424},
number = {4},
abstract = {We present an application in multi-parametrized sub-domains based
on a technique for the rapid and reliable prediction of linear-functional
outputs of elliptic coercive partial differential equations with
affine parameter dependence (reduced-basis methods). The main components
are (i) rapidly convergent global reduced-basis approximations -
Galerkin projection onto a space WN spanned by solutions of the governing
equation at N selected points in parameter space (chosen by an adaptive
procedure to minimize the estimated error and the effectivity; (ii)
a posteriori error estimation - relaxations of the error-residual
equation that provide inexpensive bounds for the error in the outputs
of interest; and (iii) off-line/on-line computational procedures
- methods which decouple the generation and projection stages of
the approximation process. The operation count for the on-line stage
- in which, given a new parameter value, we calculate the output
of interest and associated error bound - depends only on N (typically
very small) and the parametric complexity of the problem; the method
is thus ideally suited for the repeated and rapid evaluations required
in the context of parameter estimation, design, optimization, and
real-time control. The application is based on a heat transfer problem
in a parametrized geometry in view of haemodynamics applications
and biomechanical devices optimization, such as the bypass configuration
problem.},
doi = {10.1016/j.apnum.2004.11.004},
preprint = {https://infoscience.epfl.ch/record/102988/files/Rozza.pdf}
}

3. G. Rozza, "On optimization, control and shape design of an arterial bypass", International Journal for Numerical Methods in Fluids, 47(10-11), pp. 1411–1419, 2005.
Multi-level geometrical approaches in the study of aorto-coronaric bypass anastomoses configurations are discussed. The theory of optimal control based on adjoint formulation is applied in order to optimize the shape of the incoming branch of the bypass (the toe) into the coronary. At this level, two possible options are available in shape design: one implements local boundary variations in computational domain, the other, based on the theory of small perturbations, makes use of a linearized design in a reference domain. At a coarser level, reduced basis methodologies based on parametrized partial differential equations are developed to provide (a) a sensitivity analysis for geometrical quantities of interest in bypass configurations and (b) rapid and reliable prediction of integral functional outputs. The aim is (i) to provide design indications for arterial surgery in the perspective of future development for prosthetic bypasses, (ii) to develop multi-level numerical methods for optimization and shape design by optimal control, and (iii) to provide an input output relationship led by models with lower complexity and computational costs. We have numerically investigated a reduced model based on Stokes equations and a vorticity cost functional (to be minimized) in the down-field zone of bypass: a Taylor like patch has been found. A feedback procedure with Navier-Stokes fluid model is proposed based on the analysis of wall shear stress-related indexes.

@ARTICLE{Rozza2005c,
author = {Rozza, G.},
title = {On optimization, control and shape design of an arterial bypass},
journal = {International Journal for Numerical Methods in Fluids},
year = {2005},
volume = {47},
pages = {1411--1419},
number = {10-11},
abstract = {Multi-level geometrical approaches in the study of aorto-coronaric
bypass anastomoses configurations are discussed. The theory of optimal
control based on adjoint formulation is applied in order to optimize
the shape of the incoming branch of the bypass (the toe) into the
coronary. At this level, two possible options are available in shape
design: one implements local boundary variations in computational
domain, the other, based on the theory of small perturbations, makes
use of a linearized design in a reference domain. At a coarser level,
reduced basis methodologies based on parametrized partial differential
equations are developed to provide (a) a sensitivity analysis for
geometrical quantities of interest in bypass configurations and (b)
rapid and reliable prediction of integral functional outputs. The
the perspective of future development for prosthetic bypasses, (ii)
to develop multi-level numerical methods for optimization and shape
design by optimal control, and (iii) to provide an input output relationship
led by models with lower complexity and computational costs. We have
numerically investigated a reduced model based on Stokes equations
and a vorticity cost functional (to be minimized) in the down-field
zone of bypass: a Taylor like patch has been found. A feedback procedure
with Navier-Stokes fluid model is proposed based on the analysis
of wall shear stress-related indexes.},
doi = {10.1002/fld.888},
preprint = {https://infoscience.epfl.ch/record/102987/files/ICFD_J.pdf}
}

4. G. Rozza, "Real-time reduced basis techniques in arterial bypass geometries", in 3rd M.I.T. Conference on Computational Fluid and Solid Mechanics, 2005, pp. 1284–1287.
[BibTeX] [Abstract]
The reduced basis method on parametrized domains is applied to approximate blood flow through an arterial bypass. The aim is to provide (a) a sensitivity analysis for relevant geometrical quantities in bypass configurations and (b) rapid and reliable prediction of integral functional outputs (such as fluid mechanics indexes). The goal of this investigation is (i) to achieve design indications for arterial surgery in the perspective of future development for prosthetic bypasses, (ii) to develop numerical methods for optimization and design in biomechanics, and (iii) to provide an input-output relationship led by models with lower complexity and computational costs than the complete solution of fluid dynamics equations by a classical finite element method.

@INPROCEEDINGS{Rozza2005d,
author = {Rozza, G.},
title = {Real-time reduced basis techniques in arterial bypass geometries},
year = {2005},
pages = {1284--1287},
abstract = {The reduced basis method on parametrized domains is applied to approximate
blood flow through an arterial bypass. The aim is to provide (a)
a sensitivity analysis for relevant geometrical quantities in bypass
configurations and (b) rapid and reliable prediction of integral
functional outputs (such as fluid mechanics indexes). The goal of
surgery in the perspective of future development for prosthetic bypasses,
biomechanics, and (iii) to provide an input-output relationship led
by models with lower complexity and computational costs than the
complete solution of fluid dynamics equations by a classical finite
element method.},
booktitle = {3rd M.I.T. Conference on Computational Fluid and Solid Mechanics}
}

5. G. Rozza, "Real time reduced basis techniques for arterial bypass geometries", in Computational Fluid and Solid Mechanics - Third M.I.T. Conference on Computational Fluid and Solid Mechanics, 2005, pp. 1283–1287.
[BibTeX]
@INPROCEEDINGS{Rozza2005e,
author = {Rozza, Gianluigi},
title = {Real time reduced basis techniques for arterial bypass geometries},
booktitle = {Computational {F}luid and {S}olid {M}echanics - {T}hird {M}.I.T.
Conference on {C}omputational {F}luid and {S}olid {M}echanics},
year = {2005},
editor = {Bathe, K. J.},
pages = {1283--1287}
}

### 2003

1. A. Quarteroni and G. Rozza, "Optimal control and shape optimization of Aorto-Coronaric bypass anastomoses", Mathematical Models and Methods in Applied Sciences, 13(12), pp. 1801–1823, 2003.
In this paper we present a new approach in the study of Aorto-Coronaric bypass anastomoses configurations. The theory of optimal control based on adjoint formulation is applied in order to optimize the shape of the zone of the incoming branch of the bypass (the toe) into the coronary. The aim is to provide design indications in the perspective of future development for prosthetic bypasses. With a reduced model based on Stokes equations and a vorticity functional in the down field zone of bypass, a Taylor-like patch is found. A feedback procedure with Navier-Stokes fluid model is proposed based on the analysis of wall shear stress and its related indexes such as OSI.

@ARTICLE{QuarteroniRozza2003,
author = {Quarteroni, A. and Rozza, G.},
title = {Optimal control and shape optimization of Aorto-Coronaric bypass
anastomoses},
journal = {Mathematical Models and Methods in Applied Sciences},
year = {2003},
volume = {13},
pages = {1801--1823},
number = {12},
abstract = {In this paper we present a new approach in the study of Aorto-Coronaric
bypass anastomoses configurations. The theory of optimal control
based on adjoint formulation is applied in order to optimize the
shape of the zone of the incoming branch of the bypass (the toe)
into the coronary. The aim is to provide design indications in the
perspective of future development for prosthetic bypasses. With a
reduced model based on Stokes equations and a vorticity functional
in the down field zone of bypass, a Taylor-like patch is found. A
feedback procedure with Navier-Stokes fluid model is proposed based
on the analysis of wall shear stress and its related indexes such
as OSI.},
doi = {10.1142/S0218202503003124},
preprint = {https://infoscience.epfl.ch/record/102958/files/QR2003-M3AS1Rozza.pdf}
}