Block Preconditioning of Stochastic Galerkin Problems: New Two-sided Guaranteed Spectral Bounds
Abstract
The paper focuses on numerical solution of parametrized diffusion equations with scalar parameter-dependent coefficient function by the stochastic (spectral) Galerkin method. We study preconditioning of the related discretized problems using preconditioners obtained by modifying the stochastic part of the partial differential equation. We present a simple but general approach for obtaining two-sided bounds to the spectrum of the resulting matrices, based on a particular splitting of the discretized operator. Using this tool and considering the stochastic approximation space formed by classical orthogonal polynomials, we obtain new spectral bounds depending solely on the properties of the coefficient function and the type of the approximation polynomials for several classes of block-diagonal preconditioners. These bounds are guaranteed and applicable to various distributions of parameters. Moreover, the conditions on the parameter-dependent coefficient function are only local, and therefore less restrictive than those usually assumed in the literature.
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Information & Authors
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Published In
SIAM/ASA Journal on Uncertainty Quantification
Pages: 88 - 113
DOI: 10.1137/19M125902X
ISSN (online): 2166-2525
Copyright
© 2020, Society for Industrial and Applied Mathematics.
History
Submitted: 30 April 2019
Accepted: 21 October 2019
Published online: 14 January 2020
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Funding Information
Czech Academy of Sciences : L100861901
Ministry of Education, Youth and Sports of the Czech Republic : LQ1602
Grant Agency of the Czech Republic : 17-04150J
Center of Advanced Applied Sciences : CZ.02.1.01/0.0/0.0/16 019/0000778
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