Solution algorithms for stochastic galerkin discretizations of differential equations with random data

Abstract

This chapter discusses algorithms for solving systems of algebraic equations arising from stochastic Galerkin discretization of partial differential equations with random data, using the stochastic diffusion equation as a model problem. For problems in which uncertain coefficients in the differential operator are linear functions of random parameters, a variety of efficient algorithms of multigrid and multilevel type are presented, and, where possible, analytic bounds on convergence of these methods are derived. Some limitations of these approaches for problems that have nonlinear dependence on parameters are outlined, but for one example of such a problem, the diffusion equation with a diffusion coefficient that has exponential structure, a strategy is described for which the reformulated problem is also amenable to efficient solution by multigrid methods.