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

摘要

Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D is a subset of R(exp d) are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L(exp 2)(D)-orthogonal bases, and on viewing the coefficients of these expansions as random parameters y = y(omega) = (y(sub )i(omega)). This yields an equivalent parametric deterministic PDE whose solution u(x; y) is a function of both the space variable x is an element of 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 is an element of U = (- 1, 1)(exp infinity) to V = H(sup 1)(sub 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.