Tensor-Structured Galerkin Approximation of parametric and stochastic Elliptic PDEs
Boris N. Khoromskij and Christoph Schwab
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Submission date: 15. Feb. 2010
published in: SIAM journal on scientific computing, 33 (2011) 1, p. 364-385
DOI number (of the published article): 10.1137/100785715
MSC-Numbers: 65F30, 65F50, 65N35, 65F10
Keywords and phrases: elliptic operators, stochastic PDEs, the Kar\-hunen-Lo\`eve expansion, polynomial chaos, separable approximation, Kronecker-product matrix approximations, high-order tensors, preconditioners
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We investigate the convergence rate of approximations by finite sums of rank-1 tensors of solutions of multi-parametric 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 expansion.
Our approach could be regarded as either a class of compressed approximations of these solution 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 indicate that the gain from employing low-rank tensor-structured matrix formats in the numerical solution of such problems might be substantial.