NDLI: Tensor-Structured Galerkin Approximation of Parametric and Stochastic Elliptic PDEs

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Khoromskij, Boris N. Schwab, Christoph
Copyright Year	2011
Abstract	We investigate the convergence rate of approximations by finite sums of rank-1 tensors of solutions of multiparametric elliptic PDEs. Such PDEs arise, for example, in the parametric, deterministic reformulation of elliptic PDEs with random field inputs, based, for example, on the M-term truncated KarhunenLove expansion. Our approach could be regarded as either a class of compressed approximations of these solutions or as a new class of iterative elliptic problem solvers for high-dimensional, parametric, elliptic PDEs providing linear scaling complexity in the dimension M of the parameter space. It is based on rank-reduced, tensor-formatted separable approximations of the high-dimensional tensors and matrices involved in the iterative process, combined with the use of spectrally equivalent low-rank tensor-structured preconditioners to the parametric matrices resulting from a finite element discretization of the high-dimensional parametric, deterministic problems. Numerical illustrations for the M-dimensional parametric elliptic PDEs resulting from sPDEs on parameter spaces of dimensions $M\leq100$ indicate the advantages of employing low-rank tensor-structured matrix formats in the numerical solution of such problems.
Starting Page	364
Ending Page	385
Page Count	22
File Format	PDF
ISSN	10648275
DOI	10.1137/100785715
e-ISSN	10957197
Issue Number	1
Volume Number	33
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2011-02-24
Access Restriction	Subscribed
Subject Keyword	tensor-truncated iteration stochastic partial differential equations preconditioners Sparse matrices polynomial chaos elliptic operators KarhunenLove expansion Spectral, collocation and related methods Other matrix algorithms Kronecker-product matrix approximations Iterative methods for linear systems high-order tensors separable approximation
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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