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Quarteroni Alfio - One of the best experts on this subject based on the ideXlab platform.

  • MATHICSE Technical Report : Multi space reduced basis preconditioners for parametrized Stokes equations
    Écublens MATHICSE, 2019
    Co-Authors: Dal Santo Niccolò, Deparis Simone, Manzoni Andrea, Quarteroni Alfio
    Abstract:

    In this work we introduce a two-level preconditioner for the efficient solution of large scale saddlepoint linear systems arising from the finite element (FE) discretization of parametrized Stokes equations.The proposed preconditioner extends the Multi Space Reduced Basis (MSRB) preconditioning methodproposed in [12], and relies on the combination of an approximated block (fine grid) preconditioner witha reduced basis solver, which plays the role of coarse component. A sequence of RB spaces, constructedeither with an enriched velocity formulation or a Petrov-Galerkin projection, is built. As a matter offact, each RB coarse component is tailored to perform a single Iteration of the iterative method at hand.The exible GMRES (FGMRES) algorithm is employed to solve the resulting preconditioned systemand targets Small tolerances with a very Small Iteration count and in a very short time. Numerical testcases dealing with Stokes flows in three dimensional parameter-ependent geometries are consideredto assess the numerical performance of the proposed technique in different large scale computationalsettings. A detailed comparison with both the current state of the art of i) standard RB methodsand ii) preconditioning techniques for Stokes equations highlights the better efficiency of the proposedmethodology

  • MATHICSE Technical Report : Multi space reduced basis preconditioners for large-scale parametrized PDEs
    Écublens MATHICSE, 2019
    Co-Authors: Dal Santo Niccolò, Deparis Simone, Manzoni Andrea, Quarteroni Alfio
    Abstract:

    In this work we introduce a new two-level preconditioner for the efficient solution of large scale linear systems arising from the discretization of parametrized PDEs. The proposed preconditioner combines in a multiplicative way a reduced basis solver, which plays the role of coarse component, and a "traditional" fine grid preconditioner, such as one-level Additive Schwarz, block Gauss-Seidel or block Jacobi preconditioners. The coarse component is built up on a new Multi Space Reduced Basis (MSRB) method that we introduce for the first time in this paper, where are reduced basis space is built through the proper orthogonal decomposition (POD) algorithm at each step of the iterative method at hand, like the flexible GMRES method. MSRB strategy consists in building reduced basis (RB) spaces that are well-suited to perform a single Iteration, by addressing the error components which have not been treated yet. The Krylov Iterations employed to solve the resulting preconditioned system targets Small tolerances with a very Small Iteration count and in a very short time, showing good optimality and scalability properties. Simulations are carried out to evaluate the performance of the proposed preconditioner indifferent large scale computational settings related to parametrized advection diffusion equations and compared with the current state of the art algebraic multigrid preconditioners

  • Multi space reduced basis preconditioners for large-scale parametrized PDEs
    'Society for Industrial & Applied Mathematics (SIAM)', 2018
    Co-Authors: Dal Santo Niccolò, Deparis Simone, Manzoni Andrea, Quarteroni Alfio
    Abstract:

    In this work we introduce a new two-level preconditioner for the efficient solution of large-scale linear systems arising from the discretization of parametrized PDEs. The proposed preconditioner combines in a multiplicative way a reduced basis solver, which plays the role of coarse component, and a “traditional” fine-grid preconditioner, such as one-level additive Schwarz, block Gauss--Seidel, or block Jacobi preconditioners. The coarse component is built upon a new multi space reduced basis (MSRB) method that we introduce for the first time in this paper, where a reduced basis space is built through the proper orthogonal decomposition algorithm at each step of the iterative method at hand, like the flexible GMRES method. MSRB strategy consists in building reduced basis spaces that are well suited to perform a single Iteration, by addressing the error components which have not been treated yet. The Krylov Iterations employed to solve the resulting preconditioned system target Small tolerances with a very Small Iteration count and in a very short time, showing good optimality and scalability properties. Simulations are carried out to evaluate the performance of the proposed preconditioner in different large-scale computational settings related to parametrized advection diffusion equations and compared with the current state-of-the-art algebraic multigrid preconditioners

Dal Santo Niccolò - One of the best experts on this subject based on the ideXlab platform.

  • MATHICSE Technical Report : Multi space reduced basis preconditioners for parametrized Stokes equations
    Écublens MATHICSE, 2019
    Co-Authors: Dal Santo Niccolò, Deparis Simone, Manzoni Andrea, Quarteroni Alfio
    Abstract:

    In this work we introduce a two-level preconditioner for the efficient solution of large scale saddlepoint linear systems arising from the finite element (FE) discretization of parametrized Stokes equations.The proposed preconditioner extends the Multi Space Reduced Basis (MSRB) preconditioning methodproposed in [12], and relies on the combination of an approximated block (fine grid) preconditioner witha reduced basis solver, which plays the role of coarse component. A sequence of RB spaces, constructedeither with an enriched velocity formulation or a Petrov-Galerkin projection, is built. As a matter offact, each RB coarse component is tailored to perform a single Iteration of the iterative method at hand.The exible GMRES (FGMRES) algorithm is employed to solve the resulting preconditioned systemand targets Small tolerances with a very Small Iteration count and in a very short time. Numerical testcases dealing with Stokes flows in three dimensional parameter-ependent geometries are consideredto assess the numerical performance of the proposed technique in different large scale computationalsettings. A detailed comparison with both the current state of the art of i) standard RB methodsand ii) preconditioning techniques for Stokes equations highlights the better efficiency of the proposedmethodology

  • MATHICSE Technical Report : Multi space reduced basis preconditioners for large-scale parametrized PDEs
    Écublens MATHICSE, 2019
    Co-Authors: Dal Santo Niccolò, Deparis Simone, Manzoni Andrea, Quarteroni Alfio
    Abstract:

    In this work we introduce a new two-level preconditioner for the efficient solution of large scale linear systems arising from the discretization of parametrized PDEs. The proposed preconditioner combines in a multiplicative way a reduced basis solver, which plays the role of coarse component, and a "traditional" fine grid preconditioner, such as one-level Additive Schwarz, block Gauss-Seidel or block Jacobi preconditioners. The coarse component is built up on a new Multi Space Reduced Basis (MSRB) method that we introduce for the first time in this paper, where are reduced basis space is built through the proper orthogonal decomposition (POD) algorithm at each step of the iterative method at hand, like the flexible GMRES method. MSRB strategy consists in building reduced basis (RB) spaces that are well-suited to perform a single Iteration, by addressing the error components which have not been treated yet. The Krylov Iterations employed to solve the resulting preconditioned system targets Small tolerances with a very Small Iteration count and in a very short time, showing good optimality and scalability properties. Simulations are carried out to evaluate the performance of the proposed preconditioner indifferent large scale computational settings related to parametrized advection diffusion equations and compared with the current state of the art algebraic multigrid preconditioners

  • Multi space reduced basis preconditioners for large-scale parametrized PDEs
    'Society for Industrial & Applied Mathematics (SIAM)', 2018
    Co-Authors: Dal Santo Niccolò, Deparis Simone, Manzoni Andrea, Quarteroni Alfio
    Abstract:

    In this work we introduce a new two-level preconditioner for the efficient solution of large-scale linear systems arising from the discretization of parametrized PDEs. The proposed preconditioner combines in a multiplicative way a reduced basis solver, which plays the role of coarse component, and a “traditional” fine-grid preconditioner, such as one-level additive Schwarz, block Gauss--Seidel, or block Jacobi preconditioners. The coarse component is built upon a new multi space reduced basis (MSRB) method that we introduce for the first time in this paper, where a reduced basis space is built through the proper orthogonal decomposition algorithm at each step of the iterative method at hand, like the flexible GMRES method. MSRB strategy consists in building reduced basis spaces that are well suited to perform a single Iteration, by addressing the error components which have not been treated yet. The Krylov Iterations employed to solve the resulting preconditioned system target Small tolerances with a very Small Iteration count and in a very short time, showing good optimality and scalability properties. Simulations are carried out to evaluate the performance of the proposed preconditioner in different large-scale computational settings related to parametrized advection diffusion equations and compared with the current state-of-the-art algebraic multigrid preconditioners

Manzoni Andrea - One of the best experts on this subject based on the ideXlab platform.

  • MATHICSE Technical Report : Multi space reduced basis preconditioners for parametrized Stokes equations
    Écublens MATHICSE, 2019
    Co-Authors: Dal Santo Niccolò, Deparis Simone, Manzoni Andrea, Quarteroni Alfio
    Abstract:

    In this work we introduce a two-level preconditioner for the efficient solution of large scale saddlepoint linear systems arising from the finite element (FE) discretization of parametrized Stokes equations.The proposed preconditioner extends the Multi Space Reduced Basis (MSRB) preconditioning methodproposed in [12], and relies on the combination of an approximated block (fine grid) preconditioner witha reduced basis solver, which plays the role of coarse component. A sequence of RB spaces, constructedeither with an enriched velocity formulation or a Petrov-Galerkin projection, is built. As a matter offact, each RB coarse component is tailored to perform a single Iteration of the iterative method at hand.The exible GMRES (FGMRES) algorithm is employed to solve the resulting preconditioned systemand targets Small tolerances with a very Small Iteration count and in a very short time. Numerical testcases dealing with Stokes flows in three dimensional parameter-ependent geometries are consideredto assess the numerical performance of the proposed technique in different large scale computationalsettings. A detailed comparison with both the current state of the art of i) standard RB methodsand ii) preconditioning techniques for Stokes equations highlights the better efficiency of the proposedmethodology

  • MATHICSE Technical Report : Multi space reduced basis preconditioners for large-scale parametrized PDEs
    Écublens MATHICSE, 2019
    Co-Authors: Dal Santo Niccolò, Deparis Simone, Manzoni Andrea, Quarteroni Alfio
    Abstract:

    In this work we introduce a new two-level preconditioner for the efficient solution of large scale linear systems arising from the discretization of parametrized PDEs. The proposed preconditioner combines in a multiplicative way a reduced basis solver, which plays the role of coarse component, and a "traditional" fine grid preconditioner, such as one-level Additive Schwarz, block Gauss-Seidel or block Jacobi preconditioners. The coarse component is built up on a new Multi Space Reduced Basis (MSRB) method that we introduce for the first time in this paper, where are reduced basis space is built through the proper orthogonal decomposition (POD) algorithm at each step of the iterative method at hand, like the flexible GMRES method. MSRB strategy consists in building reduced basis (RB) spaces that are well-suited to perform a single Iteration, by addressing the error components which have not been treated yet. The Krylov Iterations employed to solve the resulting preconditioned system targets Small tolerances with a very Small Iteration count and in a very short time, showing good optimality and scalability properties. Simulations are carried out to evaluate the performance of the proposed preconditioner indifferent large scale computational settings related to parametrized advection diffusion equations and compared with the current state of the art algebraic multigrid preconditioners

  • Multi space reduced basis preconditioners for large-scale parametrized PDEs
    'Society for Industrial & Applied Mathematics (SIAM)', 2018
    Co-Authors: Dal Santo Niccolò, Deparis Simone, Manzoni Andrea, Quarteroni Alfio
    Abstract:

    In this work we introduce a new two-level preconditioner for the efficient solution of large-scale linear systems arising from the discretization of parametrized PDEs. The proposed preconditioner combines in a multiplicative way a reduced basis solver, which plays the role of coarse component, and a “traditional” fine-grid preconditioner, such as one-level additive Schwarz, block Gauss--Seidel, or block Jacobi preconditioners. The coarse component is built upon a new multi space reduced basis (MSRB) method that we introduce for the first time in this paper, where a reduced basis space is built through the proper orthogonal decomposition algorithm at each step of the iterative method at hand, like the flexible GMRES method. MSRB strategy consists in building reduced basis spaces that are well suited to perform a single Iteration, by addressing the error components which have not been treated yet. The Krylov Iterations employed to solve the resulting preconditioned system target Small tolerances with a very Small Iteration count and in a very short time, showing good optimality and scalability properties. Simulations are carried out to evaluate the performance of the proposed preconditioner in different large-scale computational settings related to parametrized advection diffusion equations and compared with the current state-of-the-art algebraic multigrid preconditioners

Deparis Simone - One of the best experts on this subject based on the ideXlab platform.

  • MATHICSE Technical Report : Multi space reduced basis preconditioners for parametrized Stokes equations
    Écublens MATHICSE, 2019
    Co-Authors: Dal Santo Niccolò, Deparis Simone, Manzoni Andrea, Quarteroni Alfio
    Abstract:

    In this work we introduce a two-level preconditioner for the efficient solution of large scale saddlepoint linear systems arising from the finite element (FE) discretization of parametrized Stokes equations.The proposed preconditioner extends the Multi Space Reduced Basis (MSRB) preconditioning methodproposed in [12], and relies on the combination of an approximated block (fine grid) preconditioner witha reduced basis solver, which plays the role of coarse component. A sequence of RB spaces, constructedeither with an enriched velocity formulation or a Petrov-Galerkin projection, is built. As a matter offact, each RB coarse component is tailored to perform a single Iteration of the iterative method at hand.The exible GMRES (FGMRES) algorithm is employed to solve the resulting preconditioned systemand targets Small tolerances with a very Small Iteration count and in a very short time. Numerical testcases dealing with Stokes flows in three dimensional parameter-ependent geometries are consideredto assess the numerical performance of the proposed technique in different large scale computationalsettings. A detailed comparison with both the current state of the art of i) standard RB methodsand ii) preconditioning techniques for Stokes equations highlights the better efficiency of the proposedmethodology

  • MATHICSE Technical Report : Multi space reduced basis preconditioners for large-scale parametrized PDEs
    Écublens MATHICSE, 2019
    Co-Authors: Dal Santo Niccolò, Deparis Simone, Manzoni Andrea, Quarteroni Alfio
    Abstract:

    In this work we introduce a new two-level preconditioner for the efficient solution of large scale linear systems arising from the discretization of parametrized PDEs. The proposed preconditioner combines in a multiplicative way a reduced basis solver, which plays the role of coarse component, and a "traditional" fine grid preconditioner, such as one-level Additive Schwarz, block Gauss-Seidel or block Jacobi preconditioners. The coarse component is built up on a new Multi Space Reduced Basis (MSRB) method that we introduce for the first time in this paper, where are reduced basis space is built through the proper orthogonal decomposition (POD) algorithm at each step of the iterative method at hand, like the flexible GMRES method. MSRB strategy consists in building reduced basis (RB) spaces that are well-suited to perform a single Iteration, by addressing the error components which have not been treated yet. The Krylov Iterations employed to solve the resulting preconditioned system targets Small tolerances with a very Small Iteration count and in a very short time, showing good optimality and scalability properties. Simulations are carried out to evaluate the performance of the proposed preconditioner indifferent large scale computational settings related to parametrized advection diffusion equations and compared with the current state of the art algebraic multigrid preconditioners

  • Multi space reduced basis preconditioners for large-scale parametrized PDEs
    'Society for Industrial & Applied Mathematics (SIAM)', 2018
    Co-Authors: Dal Santo Niccolò, Deparis Simone, Manzoni Andrea, Quarteroni Alfio
    Abstract:

    In this work we introduce a new two-level preconditioner for the efficient solution of large-scale linear systems arising from the discretization of parametrized PDEs. The proposed preconditioner combines in a multiplicative way a reduced basis solver, which plays the role of coarse component, and a “traditional” fine-grid preconditioner, such as one-level additive Schwarz, block Gauss--Seidel, or block Jacobi preconditioners. The coarse component is built upon a new multi space reduced basis (MSRB) method that we introduce for the first time in this paper, where a reduced basis space is built through the proper orthogonal decomposition algorithm at each step of the iterative method at hand, like the flexible GMRES method. MSRB strategy consists in building reduced basis spaces that are well suited to perform a single Iteration, by addressing the error components which have not been treated yet. The Krylov Iterations employed to solve the resulting preconditioned system target Small tolerances with a very Small Iteration count and in a very short time, showing good optimality and scalability properties. Simulations are carried out to evaluate the performance of the proposed preconditioner in different large-scale computational settings related to parametrized advection diffusion equations and compared with the current state-of-the-art algebraic multigrid preconditioners

Quarteroni A. - One of the best experts on this subject based on the ideXlab platform.

  • Multi space reduced basis preconditioners for parametrized Stokes equations
    'Elsevier BV', 2019
    Co-Authors: Dal Santo N., Deparis S., Manzoni A., Quarteroni A.
    Abstract:

    We introduce a two-level preconditioner for the efficient solution of large scale saddle point linear systems arising from the finite element (FE) discretization of parametrized Stokes equations. This preconditioner extends the Multi Space Reduced Basis (MSRB) preconditioning method proposed in Dal Santo et al. (2018); it combines an approximated block (fine grid) preconditioner with a reduced basis (RB) solver which plays the role of coarse component. A sequence of RB spaces, constructed either with an enriched velocity formulation or a Petrov - Galerkin projection, is built. Each RB coarse component is defined to perform a single Iteration of the iterative method at hand. The flexible GMRES (FGMRES) algorithm is employed to solve the resulting preconditioned system and targets Small tolerances with a very Small Iteration count and in a very short time. Numerical test cases for Stokes flows in three dimensional parameter-dependent geometries are considered to assess the numerical properties of the proposed technique in different large scale computational settings. (C) 2018 Elsevier Ltd. All rights reserved