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Tensor-Structured Galerkin Approximation of Parametric and Stochastic Elliptic PDEs

2011; Society for Industrial and Applied Mathematics; Volume: 33; Issue: 1 Linguagem: Inglês

10.1137/100785715

1095-7197

Boris N. Khoromskij, Christoph Schwab,

Advanced Numerical Methods in Computational Mathematics

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 Karhunen–Loève 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.