Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs∗
1 UPMC Univ. Paris 06, UMR 7598,
Laboratoire Jacques-Louis Lions, 75005
2 CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France.
3 Department of Mathematics, Texas A&M University, College Station, 77843 TX, USA.
4 Seminar for Applied Mathematics, ETH Zürich, 8092 Zürich, Switzerland.
Revised: 27 February 2012
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.
Mathematics Subject Classification: 65N35 / 65L10 / 35J25
Key words: Parametric and stochastic PDE’s / sparse polynomial approximation / high dimensional problems / adaptive algorithms
This research was supported by the Office of Naval Research Contracts ONR-N00014-08-1-1113, ONR N00014-09-1-0107, the AFOSR Contract FA95500910500, the ARO/DoD Contracts W911NF-05-1-0227 and W911NF-07-1-0185, the National Science Foundation Grant DMS 0915231; the excellency chair of the Foundation “Science Mathématiques de Paris” awarded to Ronald DeVore in 2009. This publication is based on work supported by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST). This research is also supported by the Swiss National Science Foundation under Grant SNF 200021-120290/1 and by the European Research Council under grant ERC AdG247277. CS acknowledges hospitality by the Hausdorff Institute for Mathematics, Bonn, Germany.
© EDP Sciences, SMAI, 2012