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ISSN Печать: 2152-5080
ISSN Онлайн: 2152-5099
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TRUNCATED HIERARCHICAL PRECONDITIONING FOR THE STOCHASTIC GALERKIN FEM
Краткое описание
Stochastic Galerkin finite element discretizations of partial differential equations with coefficients characterized by arbitrary distributions lead, in general, to fully block dense linear systems.We propose two novel strategies for constructing preconditioners for these systems to be used with Krylov subspace iterative solvers. In particular, we present a variation of the hierarchical Schur complement preconditioner, developed recently by the authors, and an adaptation of the symmetric block Gauss-Seidel method. Both preconditioners take advantage of the hierarchical structure of global stochastic Galerkin matrices, and also, when applicable, of the decay of the norms of the stiffness matrices obtained from the polynomial chaos expansion of the coefficients. This decay allows to truncate the matrix-vector multiplications in the action of the preconditioners. Also, throughout the global matrix hierarchy, we approximate solves with certain submatrices by the associated diagonal block solves. The preconditioners thus require only a limited number of stiffness matrices obtained from the polynomial chaos expansion of the coefficients, and a preconditioner for the diagonal blocks of the global matrix. The performance is illustrated by numerical experiments.
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Benner Peter, Dolgov Sergey, Onwunta Akwum, Stoll Martin, Low‐rank solution of an optimal control problem constrained by random Navier‐Stokes equations, International Journal for Numerical Methods in Fluids, 92, 11, 2020. Crossref
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Kubínová Marie, Pultarová Ivana, Block Preconditioning of Stochastic Galerkin Problems: New Two-sided Guaranteed Spectral Bounds, SIAM/ASA Journal on Uncertainty Quantification, 8, 1, 2020. Crossref
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Ghavami Mohammad S. , Sousedik Bedrich, Dabbagh Hooshang, Ahmadnasab Morad , STOCHASTIC GALERKIN FINITE ELEMENT METHOD FOR NONLINEAR ELASTICITY AND APPLICATION TO REINFORCED CONCRETE MEMBERS , International Journal for Uncertainty Quantification, 12, 6, 2022. Crossref