Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs^∗

¹ UPMC Univ. Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France.
chkifa@ann.jussieu.fr; cohen@ann.jussieu.fr
² CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France.
³ Department of Mathematics, Texas A&M University, College Station, 77843 TX, USA.
rdevore@math.tamu.ed
⁴ Seminar for Applied Mathematics, ETH Zürich, 8092 Zürich, Switzerland.
schwab@math.ethz.ch

Received: 25 June 2011
Revised: 27 February 2012

Abstract

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 = H₀¹(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.