Article

Foundations of Computational Mathematics

, Volume 10, Issue 6, pp 615-646

First online:

Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs

  • Albert CohenAffiliated withLaboratoire Jacques-Louis Lions, UMR 7598, UPMC Univ. Paris 06Laboratoire Jacques-Louis Lions, UMR 7598, CNRS Email author 
  • , Ronald DeVoreAffiliated withDepartment of Mathematics, Texas A& M University
  • , Christoph SchwabAffiliated withSeminar for Applied Mathematics, ETH Zürich

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Abstract

Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D⊂ℝ d are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L 2(D)-orthogonal bases, and on viewing the coefficients of these expansions as random parameters y=y(ω)=(y i (ω)). This yields an equivalent parametric deterministic PDE whose solution u(x,y) is a function of both the space variable xD and the in general countably many parameters y.

We establish new regularity theorems describing the smoothness properties of the solution u as a map from yU=(−1,1) to \(V=H^{1}_{0}(D)\). These results lead to analytic estimates on the V norms of the coefficients (which are functions of x) in a so-called “generalized polynomial chaos” (gpc) expansion of u.

Convergence estimates of approximations of u by best N-term truncated V valued polynomials in the variable yU are established. These estimates are of the form N r , where the rate of convergence r depends only on the decay of the random input expansion. It is shown that r exceeds the benchmark rate 1/2 afforded by Monte Carlo simulations with N “samples” (i.e., deterministic solves) under mild smoothness conditions on the random diffusion coefficients.

A class of fully discrete approximations is obtained by Galerkin approximation from a hierarchic family \(\{V_{l}\}_{l=0}^{\infty}\subset V\) of finite element spaces in D of the coefficients in the N-term truncated gpc expansions of u(x,y). In contrast to previous works, the level l of spatial resolution is adapted to the gpc coefficient. New regularity theorems describing the smoothness properties of the solution u as a map from yU=(−1,1) to a smoothness space WV are established leading to analytic estimates on the W norms of the gpc coefficients and on their space discretization error. The space W coincides with \(H^{2}(D)\cap H^{1}_{0}(D)\) in the case where D is a smooth or convex domain.

Our analysis shows that in realistic settings a convergence rate \(N_{\mathrm{dof}}^{-s}\) in terms of the total number of degrees of freedom N dof can be obtained. Here the rate s is determined by both the best N-term approximation rate r and the approximation order of the space discretization in D.

Keywords

Stochastic and parametric elliptic equations Wiener polynomial chaos Approximation rates Nonlinear approximation Sparsity

Mathematics Subject Classification (2000)

41A 65N 65C30