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Tensor-structured Galerkin Approximation of Parametric and Stochastic Elliptic PDEs
by B. N. Khoromskij and Ch. Schwab
(Report number 2010-04)
Abstract
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 dimension M<100 indicate that the gain from employing low-rank tensor-structured matrix formats in the numerical solution of such problems might be substantial.
Keywords: elliptic operators, stochastic PDEs, the Karhunen-Loève expansion, polynomial chaos, separable approximation, Kronecker-product matrix approximations, high-order tensors, preconditioners, tensor-truncated iteration
BibTeX@Techreport{KS10_58, author = {B. N. Khoromskij and Ch. Schwab}, title = {Tensor-structured Galerkin Approximation of Parametric and Stochastic Elliptic PDEs}, institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich}, number = {2010-04}, address = {Switzerland}, url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2010/2010-04.pdf }, year = {2010} }
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