# 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. 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.},
}

11. 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.}
}

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. T. Kadeethum, H. M. Nick, S. Lee, and F. Ballarin, "Enriched Galerkin Discretization for Modeling Poroelasticity and Permeability Alteration in Heterogeneous Porous Media", 2020.
Accurate simulation of the coupled fluid flow and solid deformation in porous media is challenging, especially when the media permeability and storativity are heterogeneous. We apply the enriched Galerkin (EG) finite element method for the Biot's system. Block structure used to compose the enriched space and linearization and iterative schemes employed to solve the coupled media permeability alteration are illustrated. The open-source platform used to build the block structure is presented and illustrate that it helps the enriched Galerkin method easily adaptable to any existing discontinuous Galerkin codes. Subsequently, we compare the EG method with the classic continuous Galerkin (CG) and discontinuous Galerkin (DG) finite element methods. While these methods provide similar approximations for the pressure solution of Terzaghi's one-dimensional consolidation, the CG method produces spurious oscillations in fluid pressure and volumetric strain solutions at material interfaces that have permeability contrast and does not conserve mass locally. As a result, the flux approximation of the CG method is significantly different from the one of EG and DG methods, especially for the soft materials. The difference of flux approximation between EG and DG methods is insignificant; still, the EG method demands approximately two and three times fewer degrees of freedom than the DG method for two- and three-dimensional geometries, respectively. Lastly, we illustrate that the EG method produces accurate results even for much coarser meshes.

@unpublished{KadeethumNickLeeBallarin2020b,
author = {T. Kadeethum and H.M. Nick and S. Lee and F. Ballarin},
title = {Enriched Galerkin Discretization for Modeling Poroelasticity and Permeability Alteration in Heterogeneous Porous Media},
year = {2020},
preprint = {https://arxiv.org/abs/2010.06653},
abstract = {Accurate simulation of the coupled fluid flow and solid deformation in porous media is challenging, especially when the media permeability and storativity are heterogeneous. We apply the enriched Galerkin (EG) finite element method for the Biot's system. Block structure used to compose the enriched space and linearization and iterative schemes employed to solve the coupled media permeability alteration are illustrated. The open-source platform used to build the block structure is presented and illustrate that it helps the enriched Galerkin method easily adaptable to any existing discontinuous Galerkin codes. Subsequently, we compare the EG method with the classic continuous Galerkin (CG) and discontinuous Galerkin (DG) finite element methods. While these methods provide similar approximations for the pressure solution of Terzaghi's one-dimensional consolidation, the CG method produces spurious oscillations in fluid pressure and volumetric strain solutions at material interfaces that have permeability contrast and does not conserve mass locally. As a result, the flux approximation of the CG method is significantly different from the one of EG and DG methods, especially for the soft materials. The difference of flux approximation between EG and DG methods is insignificant; still, the EG method demands approximately two and three times fewer degrees of freedom than the DG method for two- and three-dimensional geometries, respectively. Lastly, we illustrate that the EG method produces accurate results even for much coarser meshes.}
}

16. T. Kadeethum, H. M. Nick, S. Lee, and F. Ballarin, "Flow in porous media with low dimensional fractures by employing enriched Galerkin method", Advances in Water Resources, 142, pp. 103620, 2020.
[BibTeX] [Abstract] [View on publisher website]
This paper presents the enriched Galerkin discretization for modeling fluid flow in fractured porous media using the mixed-dimensional approach. The proposed method has been tested against published benchmarks. Since fracture and porous media discontinuities can significantly influence single- and multi-phase fluid flow, the heterogeneous and anisotropic matrix permeability setting is utilized to assess the enriched Galerkin performance in handling the discontinuity within the matrix domain and between the matrix and fracture domains. Our results illustrate that the enriched Galerkin method has the same advantages as the discontinuous Galerkin method; for example, it conserves local and global fluid mass, captures the pressure discontinuity, and provides the optimal error convergence rate. However, the enriched Galerkin method requires much fewer degrees of freedom than the discontinuous Galerkin method in its classical form. The pressure solutions produced by both methods are similar regardless of the conductive or non-conductive fractures or heterogeneity in matrix permeability. This analysis shows that the enriched Galerkin scheme reduces the computational costs while offering the same accuracy as the discontinuous Galerkin so that it can be applied for large-scale flow problems. Furthermore, the results of a time-dependent problem for a three-dimensional geometry reveal the value of correctly capturing the discontinuities as barriers or highly-conductive fractures.

@article{KadeethumNickLeeBallarin2020,
author = {T. Kadeethum and H.M. Nick and S. Lee and F. Ballarin},
title = {Flow in porous media with low dimensional fractures by employing enriched Galerkin method},
journal = {Advances in Water Resources},
volume = {142},
pages = {103620},
year = {2020},
abstract = {This paper presents the enriched Galerkin discretization for modeling fluid flow in fractured porous media using the mixed-dimensional approach. The proposed method has been tested against published benchmarks. Since fracture and porous media discontinuities can significantly influence single- and multi-phase fluid flow, the heterogeneous and anisotropic matrix permeability setting is utilized to assess the enriched Galerkin performance in handling the discontinuity within the matrix domain and between the matrix and fracture domains. Our results illustrate that the enriched Galerkin method has the same advantages as the discontinuous Galerkin method; for example, it conserves local and global fluid mass, captures the pressure discontinuity, and provides the optimal error convergence rate. However, the enriched Galerkin method requires much fewer degrees of freedom than the discontinuous Galerkin method in its classical form. The pressure solutions produced by both methods are similar regardless of the conductive or non-conductive fractures or heterogeneity in matrix permeability. This analysis shows that the enriched Galerkin scheme reduces the computational costs while offering the same accuracy as the discontinuous Galerkin so that it can be applied for large-scale flow problems. Furthermore, the results of a time-dependent problem for a three-dimensional geometry reveal the value of correctly capturing the discontinuities as barriers or highly-conductive fractures.}
}

17. T. Kadeethum, S. Lee, F. Ballarin, J. Choo, and H. M. Nick, "A Locally Conservative Mixed Finite Element Framework for Coupled Hydro-Mechanical-Chemical Processes in Heterogeneous Porous Media", 2020.
This paper presents a mixed finite element framework for coupled hydro-mechanical-chemical processes in heterogeneous porous media. The framework combines two types of locally conservative discretization schemes: (1) an enriched Galerkin method for reactive flow, and (2) a three-field mixed finite element method for coupled fluid flow and solid deformation. This combination ensures local mass conservation, which is critical to flow and transport in heterogeneous porous media, with a relatively affordable computational cost. A particular class of the framework is constructed for calcite precipitation/dissolution reactions, incorporating their nonlinear effects on the fluid viscosity and solid deformation. Linearization schemes and algorithms for solving the nonlinear algebraic system are also presented. Through numerical examples of various complexity, we demonstrate that the proposed framework is a robust and efficient computational method for simulation of reactive flow and transport in deformable porous media, even when the material properties are strongly heterogeneous and anisotropic.

@unpublished{KadeethumLeeBallarinChooNick2020,
author = {T. Kadeethum and S. Lee and F. Ballarin and J. Choo and H.M. Nick},
title = {A Locally Conservative Mixed Finite Element Framework for Coupled Hydro-Mechanical-Chemical Processes in Heterogeneous Porous Media},
year = {2020},
preprint = {https://arxiv.org/abs/2010.04994},
abstract = {This paper presents a mixed finite element framework for coupled hydro-mechanical-chemical processes in heterogeneous porous media. The framework combines two types of locally conservative discretization schemes: (1) an enriched Galerkin method for reactive flow, and (2) a three-field mixed finite element method for coupled fluid flow and solid deformation. This combination ensures local mass conservation, which is critical to flow and transport in heterogeneous porous media, with a relatively affordable computational cost. A particular class of the framework is constructed for calcite precipitation/dissolution reactions, incorporating their nonlinear effects on the fluid viscosity and solid deformation. Linearization schemes and algorithms for solving the nonlinear algebraic system are also presented. Through numerical examples of various complexity, we demonstrate that the proposed framework is a robust and efficient computational method for simulation of reactive flow and transport in deformable porous media, even when the material properties are strongly heterogeneous and anisotropic.}
}

18. 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},
}

19. 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. }
}

20. 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.}
}

21. 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.}
}

22. S. Perotto, M. G. Carlino, and F. Ballarin, "Model Reduction by Separation of Variables: A Comparison Between Hierarchical Model Reduction and Proper Generalized Decomposition", in Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018, 2020, pp. 61–77.
Hierarchical Model reduction and Proper Generalized Decomposition both exploit separation of variables to perform a model reduction. After setting the basics, we exemplify these techniques on some standard elliptic problems to highlight pros and cons of the two procedures, both from a methodological and a numerical viewpoint.

@InProceedings{PerottoCarlinoBallarin2020,
author = {Perotto, Simona and Carlino, Michele Giuliano and Ballarin, Francesco},
editor = {Sherwin, Spencer J. and Moxey, David and Peir{\'o}, Joaquim and Vincent, Peter E. and Schwab, Christoph},
title = {Model Reduction by Separation of Variables: A Comparison Between Hierarchical Model Reduction and Proper Generalized Decomposition},
booktitle = {Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018},
year = {2020},
publisher = {Springer International Publishing},
pages = {61--77},
preprint = {https://arxiv.org/abs/1811.11486},
doi = {10.1007/978-3-030-39647-3_4},
abstract = {Hierarchical Model reduction and Proper Generalized Decomposition both exploit separation of variables to perform a model reduction. After setting the basics, we exemplify these techniques on some standard elliptic problems to highlight pros and cons of the two procedures, both from a methodological and a numerical viewpoint.}
}

23. 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},
}

24. 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.}
}

25. 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.}
}

26. 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},
}

27. 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.}
}

28. 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},
}

29. 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.}
}

30. 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.}
}

31. 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.}
}

32. 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},
}

33. 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. 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.}
}

2. 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.}
}

3. 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}
}

4. 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}
}

5. 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}
}

6. 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.}
}

7. 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.}
}

8. 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.}
}

9. 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},
}

10. 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.}
}

11. 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.}
}

12. 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.},
}

13. 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.}
}

14. 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.}
}

15. 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.}
}

16. 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, 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}
}

2. 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},
}

3. 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}
}

4. 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}
}

5. 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. 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}
}