Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs

• Published in: ESAIM Mathematical Modelling and Numerical Analysis Vol. 47; no. 1; pp. 253 - 280
• Main Authors: Chkifa, Abdellah, Cohen, Albert, DeVore, Ronald, Schwab, Christoph
• Format: Journal Article
• Language: English
• Published: Les Ulis EDP Sciences 01.01.2013
• Subjects: 35J25; 65L10; 65N35; Adaptive algorithms; Algorithms; Applied mathematics; Approximation; Diffusion coefficient; Exact sciences and technology; high dimensional problems; Mathematical analysis; Mathematical models; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Parametric and stochastic PDE’s; Partial differential equations; Partial differential equations, boundary value problems; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Polynomials; Random variables; Sciences and techniques of general use; sparse polynomial approximation; Stochastic models; Studies; Polynomial; Convergence of numerical methods; Adaptive algorithm; Convergence acceleration; Initial value problem; Approximation algorithm; Partial differential equation; Numerical approximation; Numerical analysis; Boundary value problem; Optimal solution; Convergence rate; Hilbert space; Mathematical model; Numerical sequence
• ISSN: 0764-583X; 1290-3841; 1290-3841
• DOI: 10.1051/m2an/2012027

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.