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MiS Preprint

Tensor-Structured Galerkin Approximation of parametric and stochastic Elliptic PDEs

Boris N. Khoromskij and Christoph Schwab


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 Karhunen-Loève 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 $M\leq 100$ indicate that the gain from employing low-rank tensor-structured matrix formats in the numerical solution of such problems might be substantial.

MSC Codes:
65F30, 65F50, 65N35, 65F10
elliptic operators, stochastic PDEs, the Kar\-hunen-Lo\`eve expansion, polynomial chaos, separable approximation, Kronecker-product matrix approximations, high-order tensors, preconditioners

Related publications

2011 Repository Open Access
Boris N. Khoromskij and Christoph Schwab

Tensor-structured Galerkin approximation of parametric and stochastic elliptic PDEs

In: SIAM journal on scientific computing, 33 (2011) 1, pp. 364-385