The numerical approximation of parametric partial differential equations is a computational 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 a bounded domain D with diffusion coefficients depending on the parameters in an affine manner. For such models, it was shown in [9, 10] that under very weak assumptions on the diffusion coefficients, the entire family of solutions to such equations can be simultaneously approximated in the Hilbert space V = H_{0}^{1}(D) by multivariate sparse polynomials in the parameter vector y with a controlled number N of terms. The convergence rate in terms of N does not depend on the number of parameters in V, which may be arbitrarily large or countably infinite, thereby breaking the curse of dimensionality. However, these approximation results do not describe the concrete construction of these polynomial expansions, and should therefore rather be viewed as benchmark for the convergence analysis of numerical methods. The present paper presents an adaptive numerical algorithm for constructing a sequence of sparse polynomials that is proved to converge toward the solution with the optimal benchmark rate. Numerical experiments are presented in large parameter dimension, which confirm the effectiveness of the adaptive approach.

Keywords: parametric and stochastic PDE's, sparse polynomial approximation, high dimensional problems, adaptive algorithms

@article{M2AN_2013__47_1_253_0, author = {Chkifa, Abdellah and Cohen, Albert and DeVore, Ronald and Schwab, Christoph}, title = {Sparse adaptive {Taylor} approximation algorithms for parametric and stochastic elliptic {PDEs}}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {253--280}, publisher = {EDP-Sciences}, volume = {47}, number = {1}, year = {2013}, doi = {10.1051/m2an/2012027}, zbl = {1273.65009}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2012027/} }

TY - JOUR AU - Chkifa, Abdellah AU - Cohen, Albert AU - DeVore, Ronald AU - Schwab, Christoph TI - Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 253 EP - 280 VL - 47 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2012027/ DO - 10.1051/m2an/2012027 LA - en ID - M2AN_2013__47_1_253_0 ER -

%0 Journal Article %A Chkifa, Abdellah %A Cohen, Albert %A DeVore, Ronald %A Schwab, Christoph %T Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 253-280 %V 47 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2012027/ %R 10.1051/m2an/2012027 %G en %F M2AN_2013__47_1_253_0

Chkifa, Abdellah; Cohen, Albert; DeVore, Ronald; Schwab, Christoph. Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 (2013) no. 1, pp. 253-280. doi : 10.1051/m2an/2012027. http://www.numdam.org/articles/10.1051/m2an/2012027/

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