The paper investigates the robustness and parallel scaling properties of a novel physical factorization preconditioner with algebraic multigrid subsolves in the iterative solution of a cell-centered finite volume disc...The paper investigates the robustness and parallel scaling properties of a novel physical factorization preconditioner with algebraic multigrid subsolves in the iterative solution of a cell-centered finite volume discretization of the threedimensional multi-group radiation diffusion equations.The key idea is to take advantage of a particular kind of block factorization of the resulting system matrix and approximate the left-hand block matrix selectively spurred by parallel processing considerations.The spectral property of the preconditioned matrix is then analyzed.The practical strategy is considered sequentially and in parallel.Finally,numerical results illustrate the numerical robustness,computational efficiency and parallel strong and weak scalabilities over the real-world structured and unstructured coupled problems,showing its competitiveness with many existing block preconditioners.展开更多
The iterative solution of the sequence of linear systems arising from threetemperature(3-T)energy equations is an essential component in the numerical simulation of radiative hydrodynamic(RHD)problem.However,due to th...The iterative solution of the sequence of linear systems arising from threetemperature(3-T)energy equations is an essential component in the numerical simulation of radiative hydrodynamic(RHD)problem.However,due to the complicated application features of the RHD problems,solving 3-T linear systems with classical preconditioned iterative techniques is challenging.To address this difficulty,a physicalvariable based coarsening two-level(PCTL)preconditioner has been proposed by dividing the fully coupled system into four individual easier-to-solve subsystems.Despite its nearly optimal complexity and robustness,the PCTL algorithm suffers from poor efficiency because of the overhead associatedwith the construction of setup phase and the solution of subsystems.Furthermore,the PCTL algorithm employs a fixed strategy for solving the sequence of 3-T linear systems,which completely ignores the dynamically and slowly changing features of these linear systems.To address these problems and to efficiently solve the sequence of 3-T linear systems,we propose an adaptive two-level preconditioner based on the PCTL algorithm,referred to as αSetup-PCTL.The adaptive strategies of the αSetup-PCTL algorithm are inspired by those of αSetup-AMG algorithm,which is an adaptive-setup-based AMG solver for sequence of sparse linear systems.The proposed αSetup-PCTL algorithm could adaptively employ the appropriate strategies for each linear system,and thus increase the overall efficiency.Numerical results demonstrate that,for 36 linear systems,the αSetup-PCTL algorithm achieves an average speedup of 2.2,and a maximum speedup of 4.2 when compared to the PCTL algorithm.展开更多
We concentrate on the parallel,fully coupled and fully implicit solution of the sequence of 3-by-3 block-structured linear systems arising from the symmetrypreserving finite volume element discretization of the unstea...We concentrate on the parallel,fully coupled and fully implicit solution of the sequence of 3-by-3 block-structured linear systems arising from the symmetrypreserving finite volume element discretization of the unsteady three-temperature radiation diffusion equations in high dimensions.In this article,motivated by[M.J.Gander,S.Loisel,D.B.Szyld,SIAM J.Matrix Anal.Appl.33(2012)653–680]and[S.Nardean,M.Ferronato,A.S.Abushaikha,J.Comput.Phys.442(2021)110513],we aim to develop the additive and multiplicative Schwarz preconditioners subdividing the physical quantities rather than the underlying domain,and consider their sequential and parallel implementations using a simplified explicit decoupling factor approximation and algebraic multigrid subsolves to address such linear systems.Robustness,computational efficiencies and parallel scalabilities of the proposed approaches are numerically tested in a number of representative real-world capsule implosion benchmarks.展开更多
In this paper spectral Galerkin approximation of optimal control problem governed by fractional elliptic equation is investigated.To deal with the nonlocality of fractional Laplacian operator the Caffarelli-Silvestre ...In this paper spectral Galerkin approximation of optimal control problem governed by fractional elliptic equation is investigated.To deal with the nonlocality of fractional Laplacian operator the Caffarelli-Silvestre extension is utilized.The first order optimality condition of the extended optimal control problem is derived.A spectral Galerkin discrete scheme for the extended problem based on weighted Laguerre polynomials is developed.A priori error estimates for the spectral Galerkin discrete scheme is proved.Numerical experiments are presented to show the effectiveness of our methods and to verify the theoretical findings.展开更多
基金supported by the National Natural Science Foundation of China(Grant 11971414)Hunan National Applied Mathematics Center(Grant 2020ZYT003)the Research Foundation of Education Bureau of Hunan(Grant 21B0162).
文摘The paper investigates the robustness and parallel scaling properties of a novel physical factorization preconditioner with algebraic multigrid subsolves in the iterative solution of a cell-centered finite volume discretization of the threedimensional multi-group radiation diffusion equations.The key idea is to take advantage of a particular kind of block factorization of the resulting system matrix and approximate the left-hand block matrix selectively spurred by parallel processing considerations.The spectral property of the preconditioned matrix is then analyzed.The practical strategy is considered sequentially and in parallel.Finally,numerical results illustrate the numerical robustness,computational efficiency and parallel strong and weak scalabilities over the real-world structured and unstructured coupled problems,showing its competitiveness with many existing block preconditioners.
基金financially supported by the National Natural Science Foundation of China(62032023 and 11971414)Hunan National Applied Mathematics Center(2020ZYT003)the Research Foundation of Education Bureau of Hunan(21B0162).
文摘The iterative solution of the sequence of linear systems arising from threetemperature(3-T)energy equations is an essential component in the numerical simulation of radiative hydrodynamic(RHD)problem.However,due to the complicated application features of the RHD problems,solving 3-T linear systems with classical preconditioned iterative techniques is challenging.To address this difficulty,a physicalvariable based coarsening two-level(PCTL)preconditioner has been proposed by dividing the fully coupled system into four individual easier-to-solve subsystems.Despite its nearly optimal complexity and robustness,the PCTL algorithm suffers from poor efficiency because of the overhead associatedwith the construction of setup phase and the solution of subsystems.Furthermore,the PCTL algorithm employs a fixed strategy for solving the sequence of 3-T linear systems,which completely ignores the dynamically and slowly changing features of these linear systems.To address these problems and to efficiently solve the sequence of 3-T linear systems,we propose an adaptive two-level preconditioner based on the PCTL algorithm,referred to as αSetup-PCTL.The adaptive strategies of the αSetup-PCTL algorithm are inspired by those of αSetup-AMG algorithm,which is an adaptive-setup-based AMG solver for sequence of sparse linear systems.The proposed αSetup-PCTL algorithm could adaptively employ the appropriate strategies for each linear system,and thus increase the overall efficiency.Numerical results demonstrate that,for 36 linear systems,the αSetup-PCTL algorithm achieves an average speedup of 2.2,and a maximum speedup of 4.2 when compared to the PCTL algorithm.
基金financially supported by Hunan National Applied Mathematics Center(2020ZYT003)National Natural Science Foundation of China(11971414,62102167)+1 种基金Research Foundation of Education Bureau of Hunan(21B0162)Guangdong Basic and Applied Basic Research Foundation(2020A1515110364).
文摘We concentrate on the parallel,fully coupled and fully implicit solution of the sequence of 3-by-3 block-structured linear systems arising from the symmetrypreserving finite volume element discretization of the unsteady three-temperature radiation diffusion equations in high dimensions.In this article,motivated by[M.J.Gander,S.Loisel,D.B.Szyld,SIAM J.Matrix Anal.Appl.33(2012)653–680]and[S.Nardean,M.Ferronato,A.S.Abushaikha,J.Comput.Phys.442(2021)110513],we aim to develop the additive and multiplicative Schwarz preconditioners subdividing the physical quantities rather than the underlying domain,and consider their sequential and parallel implementations using a simplified explicit decoupling factor approximation and algebraic multigrid subsolves to address such linear systems.Robustness,computational efficiencies and parallel scalabilities of the proposed approaches are numerically tested in a number of representative real-world capsule implosion benchmarks.
基金supported by the National Natural Science Foundation of China Project(Nos.12071402,11931003,12261131501,and 11971276)the Project of Scientific Research Fund of the Hunan Provincial Science and Technology Department(No.2022RC3022).
文摘In this paper spectral Galerkin approximation of optimal control problem governed by fractional elliptic equation is investigated.To deal with the nonlocality of fractional Laplacian operator the Caffarelli-Silvestre extension is utilized.The first order optimality condition of the extended optimal control problem is derived.A spectral Galerkin discrete scheme for the extended problem based on weighted Laguerre polynomials is developed.A priori error estimates for the spectral Galerkin discrete scheme is proved.Numerical experiments are presented to show the effectiveness of our methods and to verify the theoretical findings.