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Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs*

Published online by Cambridge University Press:  28 April 2011

Christoph Schwab  and
Claude Jeffrey Gittelson
Christoph Schwab
Affiliation:
Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland E-mail: schwab@sam.math.ethz.ch
Claude Jeffrey Gittelson
Affiliation:
Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland E-mail: claude.gittelson@sam.math.ethz.ch
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Abstract

Partial differential equations (PDEs) with random input data, such as random loadings and coefficients, are reformulated as parametric, deterministic PDEs on parameter spaces of high, possibly infinite dimension. Tensorized operator equations for spatial and temporal k-point correlation functions of their random solutions are derived. Parametric, deterministic PDEs for the laws of the random solutions are derived. Representations of the random solutions' laws on infinite-dimensional parameter spaces in terms of ‘generalized polynomial chaos’ (GPC) series are established. Recent results on the regularity of solutions of these parametric PDEs are presented. Convergence rates of best N-term approximations, for adaptive stochastic Galerkin and collocation discretizations of the parametric, deterministic PDEs, are established. Sparse tensor products of hierarchical (multi-level) discretizations in physical space (and time), and GPC expansions in parameter space, are shown to converge at rates which are independent of the dimension of the parameter space. A convergence analysis of multi-level Monte Carlo (MLMC) discretizations of PDEs with random coefficients is presented. Sufficient conditions on the random inputs for superiority of sparse tensor discretizations over MLMC discretizations are established for linear elliptic, parabolic and hyperbolic PDEs with random coefficients.

Type
Research Article
Information
Acta Numerica , Volume 20 , May 2011 , pp. 291 - 467
Copyright
Copyright © Cambridge University Press 2011

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