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ISSN Печать: 2152-5080
ISSN Онлайн: 2152-5099
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BLOCK AND MULTILEVEL PRECONDITIONING FOR STOCHASTIC GALERKIN PROBLEMS WITH LOGNORMALLY DISTRIBUTED PARAMETERS AND TENSOR PRODUCT POLYNOMIALS
Краткое описание
The stochastic Galerkin method is a popular numerical method for solution of differential equations with randomly distributed data. We focus on isotropic elliptic problems with lognormally distributed coefficients. We study the block-diagonal preconditioning and the algebraic multilevel preconditioning based on the block splitting according to some hierarchy of approximation spaces for the stochastic part of the solution. We introduce upper bounds for the resulting condition numbers, and we derive a tool for obtaining sharp guaranteed upper bounds for the strengthened Cauchy-Bunyakovsky-Schwarz constant, which can serve as an indicator of the efficiency of some of these preconditioning methods. The presented multilevel approach yields a tool for efficient guaranteed two-sided a posteriori estimates of algebraic errors and for adaptive algorithms as well.
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Hrnčíř Jakub, Pultarová Ivana, Strakoš Zdeněk, Decomposition into subspaces preconditioning: abstract framework, Numerical Algorithms, 83, 1, 2020. Crossref
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Čertíková Marta, Gaynutdinova Liya, Pultarová Ivana, Multilevel a posteriori error estimator for greedy reduced basis algorithms, SN Applied Sciences, 2, 4, 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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Petrov Miroslav S., Todorov Todor D., Properties of the multidimensional finite elements, Applied Mathematics and Computation, 391, 2021. Crossref
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Bespalov Alex, Loghin Daniel, Youngnoi Rawin, Truncation Preconditioners for Stochastic Galerkin Finite Element Discretizations, SIAM Journal on Scientific Computing, 43, 5, 2021. Crossref
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Pultarová Ivana, Ladecký Martin, Two‐sided guaranteed bounds to individual eigenvalues of preconditioned finite element and finite difference problems, Numerical Linear Algebra with Applications, 28, 5, 2021. Crossref
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Plešinger Martin, Pultarová Ivana, On the extreme eigenvalues of certain matrices of non-standard inner products of Hermite polynomials, Linear Algebra and its Applications, 546, 2018. Crossref