Publications

2017

  1. F. Ballarin, E. Faggiano, A. Manzoni, A. Quarteroni, G. Rozza, S. Ippolito, C. Antona, and R. Scrofani, “Numerical modeling of hemodynamics scenarios of patient-specific coronary artery bypass grafts,” Biomechanics and Modeling in Mechanobiology, 16(4), 1373-1399, 2017.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

    A fast computational framework is devised to the study of several configurations of patient-specific coronary artery bypass grafts. This is especially useful to perform a sensitivity analysis of the haemodynamics for different flow conditions occurring in native coronary arteries and bypass grafts, the investigation of the progression of the coronary artery disease and the choice of the most appropriate surgical procedure. A complete pipeline, from the acquisition of patientspecific medical images to fast parametrized computational simulations, is proposed. Complex surgical configurations employed in the clinical practice, such as Y-grafts and sequential grafts, are studied. A virtual surgery platform based on model reduction of unsteady Navier Stokes equations for blood dynamics is proposed to carry out sensitivity analyses in a very rapid and reliable way. A specialized geometrical parametrization is employed to compare the effect of stenosis and anastomosis variation on the outcome of the surgery in several relevant cases.

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

  2. F. Ballarin, G. Rozza, and Y. Maday, “Reduced-order semi-implicit schemes for fluid-structure interaction problems,” in Model Reduction of Parametrized Systems, P. Benner, M. Ohlberger, A. Patera, G. Rozza, and K. Urban, Eds., Springer International Publishing, 2017, 149-167.
    [BibTeX] [Abstract] [View on publisher website]

    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},
    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}
    }

  3. F. Ballarin, A. D’Amario, S. Perotto, and G. Rozza, “A POD-Selective Inverse Distance Weighting method for fast parametrized shape morphing,” , 2017.
    [BibTeX] [Abstract] [Download preprint]

    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.

    @unpublished{BallarinDAmarioPerottoRozza2017,
    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 = {2017},
    preprint = {https://arxiv.org/abs/1710.09243},
    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.}
    }

  4. T. Chacón Rebollo, E. Delgado Ávila, M. Gómez Mármol, F. Ballarin, and G. Rozza, “On a certified Smagorinsky reduced basis turbulence model,” SIAM Journal on Numerical Analysis, 2017.
    [BibTeX] [Abstract] [Download preprint]

    In this work we present a reduced basis Smagorinsky turbulence model for steady flows. We approximate the non-linear eddy diffusion term using the Empirical Interpolation Method, and the velocity-pressure unknowns by an independent reduced-basis procedure. This model is based upon an a posteriori error estimation for Smagorinsky turbulence model. The theoretical development of the a posteriori error estimation is based on previous works, according to the Brezzi-Rappaz-Raviart stability theory, and adapted for the non-linear eddy diffusion term. We present some numerical tests, programmed in FreeFem++, in which we show an speedup on the computation by factor larger than 1000 in benchmark 2D flows.

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

  5. P. Chen, A. Quarteroni, and G. Rozza, “Reduced Basis Methods for Uncertainty Quantification,” SIAM/ASA Journal on Uncertainty Quantification, 5, 813-869, 2017.
    [BibTeX] [Abstract] [View on publisher website]

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

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

  6. D. Devaud and G. Rozza, “Certified Reduced Basis Method for Affinely Parametric Isogeometric Analysis NURBS Approximation,” in Spectral and High Order Methods for Partial Differential Equations, Springer, 2017, vol. 119.
    [BibTeX] [Abstract] [Download preprint]

    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}
    }

  7. 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, p. 557, 2017.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

    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},
    month = {09/2017},
    pages = {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}
    }

  8. G. Pitton and G. Rozza, “On the Application of Reduced Basis Methods to Bifurcation Problems in Incompressible Fluid Dynamics,” Journal of Scientific Computing, 2017.
    [BibTeX] [Abstract] [View on publisher website]

    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},
    author = {Giuseppe Pitton and Gianluigi Rozza}
    }

  9. G. Stabile, S. N. Hijazi, S. Lorenzi, A. Mola, and G. Rozza, “Advances in Reduced order modelling for CFD: vortex shedding around a circular cylinder using a POD-Galerkin method,” Communication in Applied Industrial Mathematics, 2017.
    [BibTeX] [Abstract] [Download preprint]

    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 = {Advances in Reduced order modelling for CFD: vortex shedding around a circular cylinder using a POD-Galerkin method},
    journal = {Communication in Applied Industrial Mathematics},
    year = {2017},
    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}
    }

  10. M. Strazzullo, F. Ballarin, R. Mosetti, and G. Rozza, “Model Reduction For Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering,” , 2017.
    [BibTeX] [Abstract] [Download preprint]

    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.

    @unpublished{StrazzulloBallarinMosettiRozza2017,
    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},
    year = {2017},
    preprint = {https://arxiv.org/abs/1710.01640},
    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, F. Salmoiraghi, A. Mola, and G. Rozza, “Dimension reduction in heterogeneous parametric spaces with application to naval engineering shape design problems,” , 2017.
    [BibTeX] [Abstract] [Download preprint]

    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.

    @unpublished{TezzeleSalmoiraghiMolaRozza2017,
    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},
    preprint = {http://arxiv.org/abs/1709.03298},
    year = {2017},
    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.},
    }

2016

  1. F. Ballarin and G. Rozza, “POD–Galerkin monolithic reduced order models for parametrized fluid-structure interaction problems,” International Journal for Numerical Methods in Fluids, 82(12), 1010-1034, 2016.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

  2. F. Ballarin, E. Faggiano, S. Ippolito, A. Manzoni, A. Quarteroni, G. Rozza, and R. Scrofani, “Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD-Galerkin method and a vascular shape parametrization,” Journal of Computational Physics, 315, 609-628, 2016.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

  3. F. Chinesta, A. Huerta, G. Rozza, and K. Willcox, “Model Order Reduction: a survey.” Wiley Encyclopedia of Computational Mechanics, 2016.
    [BibTeX] [Download preprint]
    @inbook{ChinestaHuertaRozzaWillcox2016,
    chapter = {Model Order Reduction: a survey},
    year = {2016},
    publisher = {Wiley Encyclopedia of Computational Mechanics},
    organization = {Wiley Encyclopedia of Computational Mechanics},
    author = {Francisco Chinesta and Antonio Huerta and Gianluigi Rozza and Karen Willcox},
    preprint = {https://urania.sissa.it/xmlui/bitstream/handle/1963/35194/ECM_MOR.pdf?sequence=1&isAllowed=y}
    doi = {http://eu.wiley.com/WileyCDA/WileyTitle/productCd-1119003792.html}
    }

  4. L. Iapichino, A. Quarteroni, and G. Rozza, “Reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries,” Computers and Mathematics with Applications, 71(1), 408-430, 2016.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

    @ARTICLE{IapichinoQuarteroniRozza2016,
    author = {Iapichino, L. and Quarteroni, A. and Rozza, G.},
    title = {Reduced basis method and domain decomposition for elliptic problems
    in networks and complex parametrized geometries},
    journal = {Computers and Mathematics with Applications},
    year = {2016},
    volume = {71},
    pages = {408--430},
    number = {1},
    abstract = {The aim of this work is to solve parametrized partial differential
    equations in computational domains represented by networks of repetitive
    geometries by combining reduced basis and domain decomposition techniques.
    The main idea behind this approach is to compute once, locally and
    for few reference shapes, some representative finite element solutions
    for different values of the parameters and with a set of different
    suitable boundary conditions on the boundaries: these functions will
    represent the basis of a reduced space where the global solution
    is sought for. The continuity of the latter is assured by a classical
    domain decomposition approach. Test results on Poisson problem show
    the flexibility of the proposed method in which accuracy and computational
    time may be tuned by varying the number of reduced basis functions
    employed, or the set of boundary conditions used for defining locally
    the basis functions. The proposed approach simplifies the pre-computation
    of the reduced basis space by splitting the global problem into smaller
    local subproblems. Thanks to this feature, it allows dealing with
    arbitrarily complex network and features more flexibility than a
    classical global reduced basis approximation where the topology of
    the geometry is fixed.},
    doi = {10.1016/j.camwa.2015.12.001},
    preprint = {https://www.mate.polimi.it/biblioteca/add/qmox/58-2015.pdf}
    }

  5. S. Lorenzi, A. Cammi, L. Luzzi, and G. Rozza, “POD-Galerkin Method for Finite Volume Approximation of Navier-Stokes and RANS Equations,” Computer Methods in Applied Mechanics and Engineering, 311, 151-179, 2016.
    [BibTeX] [Abstract] [View on publisher website]

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

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

  6. 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}
    }

  7. F. Salmoiraghi, F. Ballarin, L. Heltai, and G. Rozza, “Isogeometric analysis-based reduced order modelling for incompressible linear viscous flows in parametrized shapes,” Advanced Modeling and Simulation in Engineering Sciences, 3(1), p. 21, 2016.
    [BibTeX] [Abstract] [View on publisher website]

    In this work we provide a combination of isogeometric analysis with reduced order modelling techniques, based on proper orthogonal decomposition, to guarantee computational reduction for the numerical model, and with free-form deformation, for versatile geometrical parametrization. We apply it to computational fluid dynamics problems considering a Stokes flow model. The proposed reduced order model combines efficient shape deformation and accurate and stable velocity and pressure approximation for incompressible viscous flows, computed with a reduced order method. Efficient offine-online computational decomposition is guaranteed in view of repetitive calculations for parametric design and optimization problems. Numerical test cases show the efficiency and accuracy of the proposed reduced order model.

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

  8. A. Sartori, A. Cammi, L. Luzzi, and G. Rozza, “A multi-physics reduced order model for the analysis of Lead Fast Reactor single channel,” Annals of Nuclear Energy, 87, 198-208, 2016.
    [BibTeX] [Abstract] [View on publisher website]

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

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

  9. A. Sartori, A. Cammi, L. Luzzi, and G. Rozza, “Reduced basis approaches in time-dependent non-coercive settings for modelling the movement of nuclear reactor control rods,” Communications in Computational Physics, 20(1), 23-59, 2016.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

2015

  1. F. Ballarin, A. Manzoni, A. Quarteroni, and G. Rozza, “Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations,” International Journal for Numerical Methods in Engineering, 102(5), 1136-1161, 2015.
    [BibTeX] [Download preprint] [View on publisher website]
    @ARTICLE{BallarinManzoniQuarteroniRozza2015,
    author = {Ballarin, Francesco and Manzoni, Andrea and Quarteroni, Alfio and
    Rozza, Gianluigi},
    title = {Supremizer stabilization of {POD}--{G}alerkin approximation of parametrized
    steady incompressible {N}avier--{S}tokes equations},
    journal = {International Journal for Numerical Methods in Engineering},
    year = {2015},
    volume = {102},
    pages = {1136--1161},
    number = {5},
    doi = {10.1002/nme.4772},
    issn = {1097-0207},
    preprint = {https://www.mate.polimi.it/biblioteca/add/qmox/13-2014.pdf}
    }

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

  3. P. Chen, A. Quarteroni, and G. Rozza, “Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations,” Numerische Mathematik, 133(1), 67-102, 2015.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

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

  5. J. S. Hesthaven, G. Rozza, and B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations, 1 ed., Switzerland: Springer, 2015.
    [BibTeX] [Abstract] [View on publisher website]

    This book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples.

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

  6. I. Martini, G. Rozza, and B. Haasdonk, “Reduced basis approximation and a-posteriori error estimation for the coupled Stokes-Darcy system,” Advances in Computational Mathematics, 41(5), 1131-1157, 2015.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

  7. F. Negri, A. Manzoni, and G. Rozza, “Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations,” Computers and Mathematics with Applications, 69(4), 319-336, 2015.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

  8. P. Pacciarini and G. Rozza, “Reduced basis approximation of parametrized advection-diffusion PDEs with high Péclet number,” Lecture Notes in Computational Science and Engineering, 103, 419-426, 2015.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

    In this work we show some results about the reduced basis approximation of advection dominated parametrized problems, i.e. advection-diffusion problems with high Péclet number. These problems are of great importance in several engineering applications and it is well known that their numerical approximation can be affected by instability phenomena. In this work we compare two possible stabilization strategies in the framework of the reduced basis method, by showing numerical results obtained for a steady advection-diffusion problem.

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

2014

  1. F. Ballarin, A. Manzoni, G. Rozza, and S. Salsa, “Shape Optimization by Free-Form Deformation: Existence Results and Numerical Solution for Stokes Flows,” Journal of Scientific Computing, 60(3), 537-563, 2014.
    [BibTeX] [Abstract] [View on publisher website]

    Shape optimization problems governed by PDEs result from many applications in computational fluid dynamics. These problems usually entail very large computational costs and require also a suitable approach for representing and deforming efficiently the shape of the underlying geometry, as well as for computing the shape gradient of the cost functional to be minimized. Several approaches based on the displacement of a set of control points have been developed in the last decades, such as the so-called free-form deformations. In this paper we present a new theoretical result which allows to recast free-form deformations into the general class of perturbation of identity maps, and to guarantee the compactness of the set of admissible shapes. Moreover, we address both a general optimization framework based on the continuous shape gradient and a numerical procedure for solving efficiently three-dimensional optimal design problems. This framework is applied to the optimal design of immersed bodies in Stokes flows, for which we consider the numerical solution of a benchmark case study from literature.

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

  2. P. Chen, A. Quarteroni, and G. Rozza, “A weighted empirical interpolation method: A priori convergence analysis and applications,” ESAIM: Mathematical Modelling and Numerical Analysis, 48(4), 943-953, 2014.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

  3. P. Chen, A. Quarteroni, and G. Rozza, “Comparison between reduced basis and stochastic collocation methods for elliptic problems,” Journal of Scientific Computing, 59(1), 187-216, 2014.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

  4. D. Forti and G. Rozza, “Efficient geometrical parametrisation techniques of interfaces for reduced-order modelling: application to fluid–structure interaction coupling problems,” International Journal of Computational Fluid Dynamics, 28(3-4), 158-169, 2014.
    [BibTeX] [Abstract] [View on publisher website]

    We present some recent advances and improvements in shape parametrisation techniques of interfaces for reduced-order modelling with special attention to fluid–structure interaction problems and the management of structural deformations, namely, to represent them into a low-dimensional space (by control points). This allows to reduce the computational effort, and to significantly simplify the (geometrical) deformation procedure, leading to more efficient and fast reduced-order modelling applications in this kind of problems. We propose an efficient methodology to select the geometrical control points for the radial basis functions based on a modal greedy algorithm to improve the computational efficiency in view of more complex fluid–structure applications in several fields. The examples provided deal with aeronautics and wind engineering.

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

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

  6. C. Jäggli, L. Iapichino, and G. Rozza, “An improvement on geometrical parameterizations by transfinite maps,” Comptes Rendus Mathematique, 352(3), 263-268, 2014.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

    We present a method to generate a non-affine transfinite map from a given reference domain to a family of deformed domains. The map is a generalization of the Gordon-Hall transfinite interpolation approach. It is defined globally over the reference domain. Once we have computed some functions over the reference domain, the map can be generated by knowing the parametric expressions of the boundaries of the deformed domain. Being able to define a suitable map from a reference domain to a desired deformation is useful for the management of parameterized geometries.

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

  7. T. Lassila, A. Manzoni, A. Quarteroni, and G. Rozza, “Model order reduction in fluid dynamics: challenges and perspectives,” in Reduced Order Methods for Modeling and Computational Reduction, A. Quarteroni and G. Rozza, Eds., Springer MS&A Series, 2014, vol. 9, 235-274.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

    This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities – which are mainly related either to nonlinear convection terms and/or some geometric variability – that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and-in the unsteady case – long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.

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

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

  9. P. Pacciarini and G. Rozza, “Stabilized reduced basis method for parametrized scalar advection-diffusion problems at higher Péclet number: Roles of the boundary layers and inner fronts,” in 11th World Congress on Computational Mechanics, WCCM 2014, 5th European Conference on Computational Mechanics, ECCM 2014 and 6th European Conference on Computational Fluid Dynamics, ECFD 2014, 2014, 5614-5624.
    [BibTeX] [Abstract] [Download preprint]

    Advection-dominated problems, which arise in many engineering situations, often require a fast and reliable approximation of the solution given some parameters as inputs. In this work we want to investigate the coupling of the reduced basis method – which guarantees rapidity and reliability – with some classical stabilization techiques to deal with the advection-dominated condition. We provide a numerical extension of the results presented in [1], focusing in particular on problems with curved boundary layers and inner fronts whose direction depends on the parameter.

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

  10. P. Pacciarini and G. Rozza, “Stabilized reduced basis method for parametrized advection-diffusion PDEs,” Computer Methods in Applied Mechanics and Engineering, 274, 1-18, 2014.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

    In this work, we propose viable and efficient strategies for the stabilization of the reduced basis approximation of an advection dominated problem. In particular, we investigate the combination of a classic stabilization method (SUPG) with the Offline-Online structure of the RB method. We explain why the stabilization is needed in both stages and we identify, analytically and numerically, which are the drawbacks of a stabilization performed only during the construction of the reduced basis (i.e. only in the Offline stage). We carry out numerical tests to assess the performances of the “double” stabilization both in steady and unsteady problems, also related to heat transfer phenomena.

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

  11. A. Quarteroni and G. Rozza, Reduced Order Methods for Modeling and Computational Reduction, 1 ed., Springer, 2014, vol. 9.
    [BibTeX] [Abstract] [View on publisher website]

    This monograph addresses the state of the art of reduced order methods for modeling and computational reduction of complex parametrized systems, governed by ordinary and/or partial differential equations, with a special emphasis on real time computing techniques and applications in computational mechanics, bioengineering and computer graphics.</p><p>Several topics are covered, including: design, optimization, and control theory in real-time with applications in engineering; data assimilation, geometry registration, and parameter estimation with special attention to real-time computing in biomedical engineering and computational physics; real-time visualization of physics-based simulations in computer science; the treatment of high-dimensional problems in state space, physical space, or parameter space; the interactions between different model reduction and dimensionality reduction approaches; the development of general error estimation frameworks which take into account both model and discretization effects. This book is primarily addressed to computational scientists interested in computational reduction techniques for large scale differential problems.

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

  12. G. Rozza, “Fundamentals of Reduced Basis Method for problems governed by parametrized PDEs and applications,” in Separated representations and PGD-based model reduction: fundamentals and applications, Springer, 2014, vol. 554.
    [BibTeX] [Abstract] [View on publisher website]

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

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

  13. A. Sartori, D. Baroli, A. Cammi, L. Luzzi, and G. Rozza, “A reduced order model for multi-group time-dependent parametrized reactor spatial kinetics,” in International Conference on Nuclear Engineering, Proceedings, ICONE, 2014.
    [BibTeX] [Abstract] [View on publisher website]

    In this work, a Reduced Order Model (ROM) for multigroup time-dependent parametrized reactor spatial kinetics is presented. The Reduced Basis method (built upon a high-fidelity “truth” finite element approximation) has been applied to model the neutronics behavior of a parametrized system composed by a control rod surrounded by fissile material. The neutron kinetics has been described by means of a parametrized multi-group diffusion equation where the height of the control rod (i.e., how much the rod is inserted) plays the role of the varying parameter. In order to model a continuous movement of the rod, a piecewise affine transformation based on subdomain division has been implemented. The proposed ROM is capable to efficiently reproduce the neutron flux distribution allowing to take into account the spatial effects induced by the movement of the control rod with a computational speed-up of 30000 times, with respect to the “truth” model.

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

  14. A. Sartori, D. Baroli, A. Cammi, D. Chiesa, L. Luzzi, R. Ponciroli, E. Previtali, M. E. Ricotti, G. Rozza, and M. Sisti, “Comparison of a Modal Method and a Proper Orthogonal Decomposition approach for multi-group time-dependent reactor spatial kinetics,” Annals of Nuclear Energy, 71, 217-229, 2014.
    [BibTeX] [Abstract] [View on publisher website]

    In this paper, two modelling approaches based on a Modal Method (MM) and on the Proper Orthogonal Decomposition (POD) technique, for developing a control-oriented model of nuclear reactor spatial kinetics, are presented and compared. Both these methods allow developing neutronics description by means of a set of ordinary differential equations. The comparison of the outcomes provided by the two approaches focuses on the capability of evaluating the reactivity and the neutron flux shape in different reactor configurations, with reference to a TRIGA Mark II reactor. The results given by the POD-based approach are higher-fidelity with respect to the reference solution than those computed according to the MM-based approach, in particular when the perturbation concerns a reduced region of the core. If the perturbation is homogeneous throughout the core, the two approaches allow obtaining comparable accuracy results on the quantities of interest. As far as the computational burden is concerned, the POD approach ensures a better efficiency rather than direct Modal Method, thanks to the ability of performing a longer computation in the preprocessing that leads to a faster evaluation during the on-line phase.

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

2013

  1. P. Chen, A. Quarteroni, and G. Rozza, “Simulation-based uncertainty quantification of human arterial network hemodynamics,” International Journal for Numerical Methods in Biomedical Engineering, 29(6), 698-721, 2013.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

    This work aims at identifying and quantifying uncertainties from various sources in human cardiovascular system based on stochastic simulation of a one-dimensional arterial network. A general analysis of different uncertainties and probability characterization with log-normal distribution of these uncertainties is introduced. Deriving from a deterministic one-dimensional fluid-structure interaction model, we establish the stochastic model as a coupled hyperbolic system incorporated with parametric uncertainties to describe the blood flow and pressure wave propagation in the arterial network. By applying a stochastic collocation method with sparse grid technique, we study systemically the statistics and sensitivity of the solution with respect to many different uncertainties in a relatively complete arterial network with potential physiological and pathological implications for the first time.

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

  2. P. Chen, A. Quarteroni, and G. Rozza, “A weighted reduced basis method for elliptic partial differential equations with random input data,” SIAM Journal on Numerical Analysis, 51(6), 3163-3185, 2013.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

  3. P. Chen, A. Quarteroni, and G. Rozza, “Stochastic optimal robin boundary control problems of advection-dominated elliptic equations,” SIAM Journal on Numerical Analysis, 51(5), 2700-2722, 2013.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

    In this work we deal with a stochastic optimal Robin boundary control problem constrained by an advection-diffusion-reaction elliptic equation with advection-dominated term. We assume that the uncertainty comes from the advection field and consider a stochastic Robin boundary condition as control function. A stochastic saddle point system is formulated and proved to be equivalent to the first order optimality system for the optimal control problem, based on which we provide the existence and uniqueness of the optimal solution as well as some results on stochastic regularity with respect to the random variables. Stabilized finite element approximations in physical space and collocation approximations in stochastic space are applied to discretize the optimality system. A global error estimate in the product of physical space and stochastic space for the numerical approximation is derived. Illustrative numerical experiments are provided.

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

  4. D. Devaud, A. Manzoni, and G. Rozza, “A combination between the reduced basis method and the ANOVA expansion: On the computation of sensitivity indices,” Comptes Rendus Mathematique, 351(15-16), 593-598, 2013.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

  5. A. Koshakji, A. Quarteroni, and G. Rozza, “Free Form Deformation Techniques Applied to 3D Shape Optimization Problems,” Communications in Applied and Industrial Mathematics, 2013.
    [BibTeX] [Abstract] [View on publisher website]

    The purpose of this work is to analyse and study an efficient parametrization technique for a 3D shape optimization problem. After a brief review of the techniques and approaches already available in literature, we recall the Free Form Deformation parametrization, a technique which proved to be efficient and at the same time versatile, allowing to manage complex shapes even with few parameters. We tested and studied the FFD technique by establishing a path, from the geometry definition, to the method implementation, and finally to the simulation and to the optimization of the shape. In particular, we have studied a bulb and a rudder of a race sailing boat as model applications, where we have tested a complete procedure from Computer-Aided-Design to build the geometrical model to discretization and mesh generation.

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

  6. T. Lassila, A. Manzoni, and G. Rozza, “Reduction strategies for shape dependent inverse problems in haemodynamics,” IFIP Advances in Information and Communication Technology, 391 AICT, 397-406, 2013.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

  7. T. Lassila, A. Manzoni, A. Quarteroni, and G. Rozza, “Generalized reduced basis methods and n-width estimates for the approximation of the solution manifold of parametric PDEs,” in Analysis and Numerics of Partial Differential Equations, F. Brezzi, P. Colli Franzone, U. Gianazza, and G. Gilardi, Eds., , 2013, vol. 4, 307-329.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

    The set of solutions of a parameter-dependent linear partial differential equation with smooth coefficients typically forms a compact manifold in a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold We focus on operators showing an affine parametric dependence, expressed as a linear combination of parameter-independent operators through some smooth, parameter-dependent scalar functions. In the case that the parameter-dependent operator has a dominant term in its affine expansion, one can prove the existence of exponentially convergent uniform approximation spaces for the entire solution manifold These spaces can be constructed without any assumptions on the parametric regularity of the manifold – only spatial regularity of the solutions is required The exponential convergence rate is then inherited by the generalized reduced basis method We provide a numerical example related to parametrized elliptic equations confirming the predicted convergence rates.

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

  8. T. Lassila, A. Manzoni, A. Quarteroni, and G. Rozza, “Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty,” ESAIM: Mathematical Modelling and Numerical Analysis, 47(4), 1107-1131, 2013.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

  9. T. Lassila, A. Manzoni, A. Quarteroni, and G. Rozza, “A reduced computational and geometrical framework for inverse problems in hemodynamics,” International Journal for Numerical Methods in Biomedical Engineering, 29(7), 741-776, 2013.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

  10. F. Negri, G. Rozza, A. Manzoni, and A. Quarteroni, “Reduced basis method for parametrized elliptic optimal control problems,” SIAM Journal on Scientific Computing, 35(5), p. A2316–A2340, 2013.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

  11. G. Rozza, D. B. P. Huynh, and A. Manzoni, “Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: Roles of the inf-sup stability constants,” Numerische Mathematik, 125(1), 115-152, 2013.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

2012

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

  2. L. Iapichino, A. Quarteroni, and G. Rozza, “A reduced basis hybrid method for the coupling of parametrized domains represented by fluidic networks,” Computer Methods in Applied Mechanics and Engineering, 221-222, 63-82, 2012.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

  3. T. Lassila, A. Quarteroni, and G. Rozza, “A reduced basis model with parametric coupling for fluid-structure interaction problems,” SIAM Journal on Scientific Computing, 34(2), p. A1187–A1213, 2012.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

    We present a new model reduction technique for steady fluid-structure interaction problems. When the fluid domain deformation is suitably parametrized, the coupling conditions between the fluid and the structure can be formulated in the low-dimensional space of geometric parameters. Moreover, we apply the reduced basis method to reduce the cost of repeated fluid solutions necessary to achieve convergence of fluid-structure iterations. In this way a reduced order model with reliable a posteriori error bounds is obtained. The proposed method is validated with an example of steady Stokes flow in an axisymmetric channel, where the structure is described by a simple one-dimensional generalized string model. We demonstrate rapid convergence of the reduced solution of the parametrically coupled problem as the number of geometric parameters is increased.

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

  4. T. Lassila, A. Manzoni, and G. Rozza, “On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition,” ESAIM: Mathematical Modelling and Numerical Analysis, 46(6), 1555-1576, 2012.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

    A new approach for computationally efficient estimation of stability factors for parametric partial differential equations is presented. The general parametric bilinear form of the problem is approximated by two affinely parametrized bilinear forms at different levels of accuracy (after an empirical interpolation procedure). The successive constraint method is applied on the coarse level to obtain a lower bound for the stability factors, and this bound is extended to the fine level by adding a proper correction term. Because the approximate problems are affine, an efficient offline/online computational scheme can be developed for the certified solution (error bounds and stability factors) of the parametric equations considered. We experiment with different correction terms suited for a posteriori error estimation of the reduced basis solution of elliptic coercive and noncoercive problems.

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

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

  6. A. Manzoni, A. Quarteroni, and G. Rozza, “Model reduction techniques for fast blood flow simulation in parametrized geometries,” International Journal for Numerical Methods in Biomedical Engineering, 28(6-7), 604-625, 2012.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

  7. A. Manzoni, A. Quarteroni, and G. Rozza, “Computational Reduction for Parametrized PDEs: Strategies and Applications,” Milan Journal of Mathematics, 80(2), 283-309, 2012.
    [BibTeX] [Abstract] [View on publisher website]

    In this paper we present a compact review on the mostly used techniques for computational reduction in numerical approximation of partial differential equations. We highlight the common features of these techniques and provide a detailed presentation of the reduced basis method, focusing on greedy algorithms for the construction of the reduced spaces. An alternative family of reduction techniques based on surrogate response surface models is briefly recalled too. Then, a simple example dealing with inviscid flows is presented, showing the reliability of the reduced basis method and a comparison between this technique and some surrogate models.

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

  8. A. Manzoni, A. Quarteroni, and G. Rozza, “Shape optimization for viscous flows by reduced basis methods and free-form deformation,” International Journal for Numerical Methods in Fluids, 70(5), 646-670, 2012.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

    @ARTICLE{ManzoniQuarteroniRozza2012,
    author = {Manzoni, A. and Quarteroni, A. and Rozza, G.},
    title = {Shape optimization for viscous flows by reduced basis methods and
    free-form deformation},
    journal = {International Journal for Numerical Methods in Fluids},
    year = {2012},
    volume = {70},
    pages = {646--670},
    number = {5},
    abstract = {In this paper, we further develop an approach previously introduced
    in Lassila and Rozza, 2010, for shape optimization that combines
    a suitable low-dimensional parametrization of the geometry (yielding
    a geometrical reduction) with reduced basis methods (yielding a reduction
    of computational complexity). More precisely, free-form deformation
    techniques are considered for the geometry description and its parametrization,
    whereas reduced basis methods are used upon a FE discretization to
    solve systems of parametrized partial differential equations. This
    allows an efficient flow field computation and cost functional evaluation
    during the iterative optimization procedure, resulting in effective
    computational savings with respect to usual shape optimization strategies.
    This approach is very general and can be applied to a broad variety
    of problems. In this paper, we apply it to find the optimal shape
    of aorto-coronaric bypass anastomoses based on vorticity minimization
    in the down-field region. Blood flows in the coronary arteries are
    modeled using Stokes equations; afterwards, results have been verified
    in feedback using Navier-Stokes equations.},
    doi = {10.1002/fld.2712},
    preprint = {https://www.mate.polimi.it/biblioteca/add/qmox/31-2010.pdf}
    }

  9. G. Rozza, A. Manzoni, and F. Negri, “Reduction strategies for PDE-constrained optimization problems in haemodynamics,” in ECCOMAS 2012 – European Congress on Computational Methods in Applied Sciences and Engineering, 2012, 1749-1768.
    [BibTeX] [Abstract] [Download preprint]

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

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

2011

  1. F. Gelsomino and G. Rozza, “Comparison and combination of reduced-order modelling techniques in 3D parametrized heat transfer problems,” Mathematical and Computer Modelling of Dynamical Systems, 17(4), 371-394, 2011.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

    Reduced basis method has successfully been used in 2D to solve heat transfer parametrized problems. In this work, we present some 3D applications in the same field.We consider two problems, the steady Thermal Fin and the time-dependent Graetz Flow, we compare two reduced-order modelling techniques: Reduced basis and Proper orthogonal decomposition, then we apply a combination of the two strategies in the time-dependent case.

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

  2. T. Lassila and G. Rozza, “Model reduction of semiaffinely parameterized partial differential equations by two-level affine approximation,” Comptes Rendus Mathematique, 349(1-2), 61-66, 2011.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

    We propose an improvement to the reduced basis method for parametric partial differential equations. An assumption of affine parameterization leads to an efficient offline-online decomposition when the problem is solved for many different parametric configurations. We consider an advection-diffusion problem, where the diffusive term is non-affinely parameterized and treated with a two-level affine approximation given by the empirical interpolation method. The offline stage and a posteriori error estimation is performed using the coarse-level approximation, while the fine-level approximation is used to perform a correction iteration that reduces the actual error of the reduced basis approximation while keeping the same certified error bounds.

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

  3. A. Quarteroni, G. Rozza, and A. Manzoni, “Certified reduced basis approximation for parametrized partial differential equations and applications,” Journal of Mathematics in Industry, 1(1), 1-49, 2011.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

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

  5. G. Rozza, T. Lassila, and A. Manzoni, “Reduced basis approximation for shape optimization in thermal flows with a parametrized polynomial geometric map,” Lecture Notes in Computational Science and Engineering, 76, 307-315, 2011.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

    Reduced basis approximations for geometrically parametrized advection-diffusion equations are investigated. The parametric domains are assumed to be images of a reference domain through a piecewise polynomial map; this may lead to nonaffinely parametrized diffusion tensors that are treated with an empirical interpolation method. An a posteriori error bound including a correction term due to this approximation is given. Results concerning the applied methodology and the rigor of the corrected error estimator are shown for a shape optimization problem in a thermal flow.

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

  6. G. Rozza, “Reduced basis approximation and error bounds for potential flows in parametrized geometries,” Communications in Computational Physics, 9(1), 1-48, 2011.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

2010

  1. T. M. Lassila and G. Rozza, “Reduced formulation of a steady fluid-structure interaction problem with parametric coupling,” in Proceedings of the 10th Finnish Mechanics Days Conference, 2010.
    [BibTeX] [Abstract]

    We propose a two-fold approach to model reduction of fluid-structure interaction. The state equations for the fluid are solved with reduced basis methods. These are model reduction methods for parametric partial differential equations using well-chosen snapshot solutions in order to build a set of global basis functions. The other reduction is in terms of the geometric complexity of the moving fluidstructure interface. We use free-form deformations to parameterize the perturbation of the flow channel at rest configuration. As a computational example we consider a steady fluid-structure interaction problem: an incompressible Stokes flow in a channel that has a flexible wall.

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

  2. T. Lassila and G. Rozza, “Parametric free-form shape design with PDE models and reduced basis method,” Computer Methods in Applied Mechanics and Engineering, 199(23-24), 1583-1592, 2010.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

    We present a coupling of the reduced basis methods and free-form deformations for shape optimization and design of systems modelled by elliptic PDEs. The free-form deformations give a parameterization of the shape that is independent of the mesh, the initial geometry, and the underlying PDE model. The resulting parametric PDEs are solved by reduced basis methods. An important role in our implementation is played by the recently proposed empirical interpolation method, which allows approximating the non-affinely parameterized deformations with affinely parameterized ones. These ingredients together give rise to an efficient online computational procedure for a repeated evaluation design environment like the one for shape optimization. The proposed approach is demonstrated on an airfoil inverse design problem.

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

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

  4. G. Rozza and A. Manzoni, “Model Order Reduction by geometrical parametrization for shape optimization in computational fluid dynamics,” in Proceedings of the ECCOMAS CFD 2010, V European Conference on Computational Fluid Dynamics, 2010.
    [BibTeX] [Abstract]

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

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

  5. Z. C. Xuan, T. Lassila, G. Rozza, and A. Quarteroni, “On computing upper and lower bounds on the outputs of linear elasticity problems approximated by the smoothed finite element method,” International Journal for Numerical Methods in Engineering, 83(2), 174-195, 2010.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

2009

  1. S. Deparis and G. Rozza, “Reduced basis method for multi-parameter-dependent steady Navier-Stokes equations: Applications to natural convection in a cavity,” Journal of Computational Physics, 228(12), 4359-4378, 2009.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

  2. N. -C. Nguyen, G. Rozza, and A. T. Patera, “Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation,” Calcolo, 46(3), 157-185, 2009.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

  3. G. Rozza, C. N. Nguyen, A. T. Patera, and S. Deparis, “Reduced basis methods and a posteriori error estimators for heat transfer problems,” in Proceedings of the ASME Summer Heat Transfer Conference 2009, HT2009, 2009, 753-762.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

  4. G. Rozza and A. Quarteroni, “Reduced basis approximation for parametrized partial differential equations,” in Numerical Models for Differential Equations, E. Zuazua, C. Le Bris, A. T. Patera, A. Quarteroni, and T. Hou, Eds., Springer, 2009, vol. 2, 556-587.
    [BibTeX] [Abstract]

    In this chapter we describe the basic ideas of reduced basis (RB) approximation methods for rapid and reliable evaluation of input-output relationships in which the output is expressed as a functional of a field variable that is the solution of an input-parametrized partial differential equation (PDE). We shall focus on linear output functionals and affinely parametrized linear elliptic coercive PDEs.

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

  5. G. Rozza, “An introduction to reduced basis method for parametrized PDEs,” in Applied and Industrial Mathematics in Italy, 2009, 508-519.
    [BibTeX] [Abstract] [Download preprint]

    We provide an introduction on reduced basis (RB) method for the solution of parameterized partial differential equations (PDEs). We introduce all the main ingredients to describe the methodology and the algorithms used to build the approximation spaces and the error bounds. We consider a model problem describing a steady potential flow around parametrized bodies and we provide some illustrative results.

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

  6. G. Rozza, “Reduced basis methods for Stokes equations in domains with non-affine parameter dependence,” Computing and Visualization in Science, 12(1), 23-35, 2009.
    [BibTeX] [Abstract] [View on publisher website]

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

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

  7. G. Rozza, D. B. P. Huynh, N. C. Nguyen, and A. T. Patera, “Real-time reliable simulation of heat transfer phenomena,” in Proceedings of the ASME Summer Heat Transfer Conference 2009, HT2009, 2009, 851-860.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

    In this paper we discuss the application of the certified reduced basis method and the associated software package rbMIT to “worked problems” in steady and unsteady conduction. Each worked problem is characterized by an input parameter vector – material properties, boundary conditions and sources, and geometry – and desired outputs – selected fluxes and temperatures. The methodology and associated rbMIT software, as well as the educational worked problem framework, consists of two distinct stages: an Offline (or “Instructor”) stage in which a new heat transfer worked problem is first created; and an Online (or “Lecturer”/ “Student”) stage in which the worked problem is subsequently invoked in (say) various inclass, project, or homework settings. In the very inexpensive Online stage, given an input parameter value, the software returns both (i) an accurate reduced basis output prediction, and (ii) a rigorous bound for the error in the reduced basis prediction relative to an underlying expensive high-fidelity finite element discretization; as required in the educational context, the response is both rapid and reliable. We present illustrative results for two worked problems: a steady thermal fin, and unsteady thermal analysis of a delamination crack.

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

2008

  1. B. Haasdonk, M. Ohlberger, and G. Rozza, “A reduced basis method for evolution schemes with parameter-dependent explicit operators,” Electronic Transactions on Numerical Analysis, 32, 145-161, 2008.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

  2. R. Milani, A. Quarteroni, and G. Rozza, “Reduced basis method for linear elasticity problems with many parameters,” Computer Methods in Applied Mechanics and Engineering, 197(51-52), 4812-4829, 2008.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

  3. G. Rozza, D. B. P. Huynh, and A. T. Patera, “Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: Application to transport and continuum mechanics,” Archives of Computational Methods in Engineering, 15(3), 229-275, 2008.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

    In this paper we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equations. The essential ingredients are (primal-dual) Galerkin projection onto a low-dimensional space associated with a smooth “parametric manifold”-dimension reduction; efficient and effective greedy sampling methods for identification of optimal and numerically stable approximations-rapid convergence; a posteriori error estimation procedures-rigorous and sharp bounds for the linear-functional outputs of interest; and Offline-Online computational decomposition strategies-minimum marginal cost for high performance in the real-time/embedded (e.g., parameter-estimation, control) and many-query (e.g., design optimization, multi-model/scale) contexts. We present illustrative results for heat conduction and convection-diffusion, inviscid flow, and linear elasticity; outputs include transport rates, added mass, and stress intensity factors.

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

2007

  1. D. B. P. Huynh, G. Rozza, S. Sen, and A. T. Patera, “A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants,” Comptes Rendus Mathematique, 345(8), 473-478, 2007.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

  2. A. Quarteroni and G. Rozza, “Numerical solution of parametrized Navier-Stokes equations by reduced basis methods,” Numerical Methods for Partial Differential Equations, 23(4), 923-948, 2007.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

    @ARTICLE{QuarteroniRozza2007,
    author = {Quarteroni, A. and Rozza, G.},
    title = {Numerical solution of parametrized {N}avier-{S}tokes equations by
    reduced basis methods},
    journal = {Numerical Methods for Partial Differential Equations},
    year = {2007},
    volume = {23},
    pages = {923--948},
    number = {4},
    abstract = {We apply the reduced basis method to solve Navier-Stokes equations
    in parametrized domains. Special attention is devoted to the treatment
    of the parametrized nonlinear transport term in the reduced basis
    framework, including the case of nonaffine parametric dependence
    that is treated by an empirical interpolation method. This method
    features (i) a rapid global convergence owing to the property of
    the Galerkin projection onto a space WN spanned by solutions of the
    governing partial differential equation at N (optimally) selected
    points in the parameter space, and (ii) the offline/online computational
    procedures that decouple the generation and projection stages of
    the approximation process. This method is well suited for the repeated
    and rapid evaluations required in the context of parameter estimation,
    design, optimization, and real-time control. Our analysis focuses
    on: (i) the pressure treatment of incompressible Navier-Stokes problem;
    (ii) the fulfillment of an equivalent inf-sup condition to guarantee
    the stability of the reduced basis solutions. The applications that
    we consider involve parametrized geometries, like e.g. a channel
    with curved upper wall or an arterial bypass configuration.},
    doi = {10.1002/num.20249},
    preprint = {https://www.mate.polimi.it/biblioteca/add/qmox/mox88.pdf}
    }

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

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

  5. G. Rozza and K. Veroy, “On the stability of the reduced basis method for Stokes equations in parametrized domains,” Computer Methods in Applied Mechanics and Engineering, 196(7), 1244-1260, 2007.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

2006

  1. V. Agoshkov, A. Quarteroni, and G. Rozza, “A mathematical approach in the design of arterial bypass using unsteady Stokes equations,” Journal of Scientific Computing, 28(2-3), 139-165, 2006.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

    In this paper we present an approach for the study of Aorto-Coronaric bypass anastomoses configurations using unsteady Stokes equations. The theory of optimal control based on adjoint formulation is applied in order to optimize the shape of the zone of the incoming branch of the bypass (the toe) into the coronary according to several optimality criteria.

    @ARTICLE{AgoshkovQuarteroniRozza2006a,
    author = {Agoshkov, V. and Quarteroni, A. and Rozza, G.},
    title = {A mathematical approach in the design of arterial bypass using unsteady
    {S}tokes equations},
    journal = {Journal of Scientific Computing},
    year = {2006},
    volume = {28},
    pages = {139--165},
    number = {2-3},
    abstract = {In this paper we present an approach for the study of Aorto-Coronaric
    bypass anastomoses configurations using unsteady Stokes equations.
    The theory of optimal control based on adjoint formulation is applied
    in order to optimize the shape of the zone of the incoming branch
    of the bypass (the toe) into the coronary according to several optimality
    criteria.},
    doi = {10.1007/s10915-006-9077-9},
    preprint = {https://www.mate.polimi.it/biblioteca/add/qmox/mox54.pdf}
    }

  2. V. Agoshkov, A. Quarteroni, and G. Rozza, “Shape design in aorto-coronaric bypass anastomoses using perturbation theory,” SIAM Journal on Numerical Analysis, 44(1), 367-384, 2006.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

    In this paper we present a new approach in the study of aorto-coronaric bypass anastomoses configurations based on small perturbation theory. The theory of optimal control based on adjoint formulation is applied in order to optimize the shape of the zone of the incoming branch of the bypass (the toe) into the coronary (see Figure 2.1). The aim is to provide design indications in the perspective of future development for prosthetic bypasses.

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

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

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

  5. A. Quarteroni, G. Rozza, L. Dedè, and A. Quaini, “Numerical approximation of a control problem for advection-diffusion processes,” IFIP International Federation for Information Processing, 199, 261-273, 2006.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

    Two different approaches are proposed to enhance the efficiency of the numerical resolution of optimal control problems governed by a linear advection-diffusion equation. In the framework of the Galerkin-Finite Element (FE) method, we adopt a novel a posteriori error estimate of the discretization error on the cost functional; this estimate is used in the course of a numerical adaptive strategy for the generation of efficient grids for the resolution of the optimal control problem. Moreover, we propose to solve the control problem by adopting a reduced basis (RB) technique, hence ensuring rapid, reliable and repeated evaluations of input-output relationship. Our numerical tests show that by this technique a substantial saving of computational costs can be achieved.

    @ARTICLE{QuarteroniRozzaDedeQuaini2006,
    author = {Quarteroni, A. and Rozza, G. and Ded\`e, L. and Quaini, A.},
    title = {Numerical approximation of a control problem for advection-diffusion
    processes},
    journal = {IFIP International Federation for Information Processing},
    year = {2006},
    volume = {199},
    pages = {261--273},
    abstract = {Two different approaches are proposed to enhance the efficiency of
    the numerical resolution of optimal control problems governed by
    a linear advection-diffusion equation. In the framework of the Galerkin-Finite
    Element (FE) method, we adopt a novel a posteriori error estimate
    of the discretization error on the cost functional; this estimate
    is used in the course of a numerical adaptive strategy for the generation
    of efficient grids for the resolution of the optimal control problem.
    Moreover, we propose to solve the control problem by adopting a reduced
    basis (RB) technique, hence ensuring rapid, reliable and repeated
    evaluations of input-output relationship. Our numerical tests show
    that by this technique a substantial saving of computational costs
    can be achieved.},
    doi = {10.1007/0-387-33006-2_24},
    year = {2006},
    preprint = {https://www.mate.polimi.it/biblioteca/add/qmox/mox69.pdf}
    }

  6. G. Rozza, “Real-time reduced basis solutions for Navier-Stokes equations: optimization of parametrized bypass configurations,” in ECCOMAS CFD 2006 Proceedings on CFD, 2006, 1-16.
    [BibTeX] [Abstract] [Download preprint]

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

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

2005

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

  2. G. Rozza, “Real-time reduced basis techniques in arterial bypass geometries,” in 3rd M.I.T. Conference on Computational Fluid and Solid Mechanics, 2005, 1284-1287.
    [BibTeX] [Abstract]

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

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

  3. G. Rozza, “On optimization, control and shape design of an arterial bypass,” International Journal for Numerical Methods in Fluids, 47(10-11), 1411-1419, 2005.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

  4. G. Rozza, “Reduced-basis methods for elliptic equations in sub-domains with a posteriori error bounds and adaptivity,” Applied Numerical Mathematics, 55(4), 403-424, 2005.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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

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

2003

  1. A. Quarteroni and G. Rozza, “Optimal control and shape optimization of Aorto-Coronaric bypass anastomoses,” Mathematical Models and Methods in Applied Sciences, 13(12), 1801-1823, 2003.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]

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

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