An adaptive stochastic Galerkin method for random elliptic operators

We derive an adaptive solver for random elliptic boundary value problems, using techniques from adaptive wavelet methods. Substituting wavelets by polynomials of the random parameters leads to a modular solver for the parameter dependence of the random solution, which combines with any discretization on the spatial domain. In addition to selecting active polynomial modes, this solver can adaptively construct a separate spatial discretization for each of their coefficients. We show convergence of the solver in this general setting, along with a computable bound for the mean square error, and an optimality property in the case of a single spatial discretization. Numerical computations demonstrate convergence of the solver and compare it to a sparse tensor product construction. Introduction Stochastic Galerkin methods have emerged in the past decade as an efficient solution procedure for boundary value problems depending on random data; see [14, 32, 2, 30, 23, 18, 31, 28, 6, 5]. These methods approximate the random solution by a Galerkin projection onto a finite-dimensional space of random fields. This requires the solution of a single coupled system of deterministic equations for the coefficients of the Galerkin projection with respect to a predefined set of basis functions on the parameter domain. A major remaining obstacle is the construction of suitable spaces in which to compute approximate solutions. These should be adapted to the stochastic structure of the equation. Simple tensor product constructions are infeasible due to the high dimensionality of the parameter domain in the case of input random fields with low regularity. Parallel to but independently from the development of stochastic Galerkin methods, a new class of adaptive methods has emerged, which are set not in the continuous framework of a boundary value problem, but rather on the level of coefficients with respect to a hierarchic Riesz basis, such as a wavelet basis. Due to the norm equivalences constitutive of Riesz bases, errors and residuals in appropriate sequence spaces are equivalent to those in physically meaningful function spaces. This permits adaptive wavelet methods to be applied directly to a large class of equations, provided that a suitable Riesz basis is available. Received by the editor March 2, 2011 and, in revised form, September 24, 2011. 2010 Mathematics Subject Classification. Primary 35R60, 47B80, 60H25, 65C20, 65N12, 65N22, 65N30, 65J10, 65Y20. This research was supported in part by the Swiss National Science Foundation grant No. 200021-120290/1. c ©2013 American Mathematical Society Reverts to public domain 28 years from publication