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

• Published in: Foundations of computational mathematics Vol. 10; no. 6; pp. 615 - 646
• Main Authors: Cohen, Albert, DeVore, Ronald, Schwab, Christoph
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.12.2010; Springer; Springer Nature B.V
• Subjects: Analysis; Applications of Mathematics; Approximation; Computational mathematics; Computer Science; Computer simulation; Convergence; Economics; Estimates; Galerkin methods; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Matrix Theory; Monte Carlo method; Numerical Analysis; Smoothness; Theorems; Wiener polynomial chaos; 65C30; Stochastic and parametric elliptic equations; Sparsity; 41A; Approximation rates; Nonlinear approximation; 65N
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-010-9072-2

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 x ∈ D 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 y ∈ U =(−1,1) ∞ to . 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 y ∈ U 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 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 y ∈ U =(−1,1) ∞ to a smoothness space W ⊂ V 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 in the case where D is a smooth or convex domain. Our analysis shows that in realistic settings a convergence rate 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 .