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

Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D ⊂ R are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L(D)-orthogonal bases, and on viewing the coefficients of these expansions as random parameters y = y(ω) = (yi(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 decribing the smoothness properties of the solution u as a map from y ∈ U = (−1, 1)∞ to V = H 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 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 {Vl}l=0 ⊂ 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 decribing 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 H(D) ∩ H 0 (D) in the case where D is a smooth or convex domain. Our analysis shows that in realistic settings a convergence rate N−s d.o.f in terms of the total number of degrees of freedom Nd.o.f 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. ∗This research was supported by the Fondation Sciences Mathématiques de Paris; the Office of Naval Research Contracts ONR-N00014-08-1-1113, ONR N00014-09-1-0107; the AFOSR Contract FA95500910500; the NSF Grant DMS-0810869; the Swiss National Science Foundation under Grant No. 200021-120290/1 and European Research Council Project No. 247277.