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Sparse adaptive Taylor approximation algorithms for parametricand stochastic elliptic PDEs∗

Published online by Cambridge University Press:  29 November 2012

Abdellah Chkifa ,
Albert Cohen ,
Ronald DeVore  and
Christoph Schwab
Abdellah Chkifa
Affiliation:
UPMC Univ. Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France.. chkifa@ann.jussieu.fr; cohen@ann.jussieu.fr CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France.
Albert Cohen
Affiliation:
UPMC Univ. Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France.. chkifa@ann.jussieu.fr; cohen@ann.jussieu.fr CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France.
Ronald DeVore
Affiliation:
Department of Mathematics, Texas A&M University, College Station, 77843 TX, USA.; rdevore@math.tamu.ed
Christoph Schwab
Affiliation:
Seminar for Applied Mathematics, ETH Zürich, 8092 Zürich, Switzerland.; schwab@math.ethz.ch
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Abstract

The numerical approximation of parametric partial differential equations is acomputational challenge, in particular when the number of involved parameter is large.This paper considers a model class of second order, linear, parametric, elliptic PDEs on abounded domain D with diffusion coefficients depending on the parametersin an affine manner. For such models, it was shown in [9, 10] that under very weak assumptionson the diffusion coefficients, the entire family of solutions to such equations can besimultaneously approximated in the Hilbert space V = H01(D) by multivariate sparse polynomials in the parametervector y with a controlled number N of terms. Theconvergence rate in terms of N does not depend on the number ofparameters in V, which may be arbitrarily large or countably infinite,thereby breaking the curse of dimensionality. However, these approximation results do notdescribe the concrete construction of these polynomial expansions, and should thereforerather be viewed as benchmark for the convergence analysis of numerical methods. Thepresent paper presents an adaptive numerical algorithm for constructing a sequence ofsparse polynomials that is proved to converge toward the solution with the optimalbenchmark rate. Numerical experiments are presented in large parameter dimension, whichconfirm the effectiveness of the adaptive approach.

Keywords

Parametric and stochastic PDE’ssparse polynomial approximationhigh dimensional problemsadaptive algorithms
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 1 , January 2013 , pp. 253 - 280
Copyright
© EDP Sciences, SMAI, 2012

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