Journal
SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION
Volume 8, Issue 1, Pages 88-113Publisher
SIAM PUBLICATIONS
DOI: 10.1137/19M125902X
Keywords
stochastic Galerkin method; preconditioning; block-diagonal preconditioning; spectral bounds; diffusion problem
Funding
- 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
- European Regional Development Fund [CZ.02.1.01/0.0/0.0/16 019/0000778]
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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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