Preconditioning Stochastic Galerkin Saddle Point Systems

Abstract

Abstract. Mixed finite element discretizations of deterministic second-order elliptic partial differential equations (PDEs) lead to saddle point systems for which the study of iterative solvers and preconditioners is mature. Galerkin approximation of solutions of stochastic second-order elliptic PDEs, which couple standard mixed finite element discretizations in physical space with global polynomial approximation on a probability space, also give rise to linear systems with familiar saddle point structure. For stochastically nonlinear problems, the solution of such systems presents a serious computational challenge. The blocks are sums of Kronecker products of pairs of matrices associated with two distinct discretizations and the systems are large, reflecting the curse of dimensionality inherent in most stochastic approximation schemes. Moreover, for the problems considered herein, the leading blocks of the saddle point matrices are block-dense and the cost of a matrix vector product is non-trivial.We implement a stochastic Galerkin discretization for the steady-state diffusion problem written as a mixed firstorder system. The diffusion coefficient is assumed to be a lognormal random field, approximated via a nonlinear function of a finite number of unbounded random parameters. We study the resulting saddle point systems and investigate the efficiency of block-diagonal preconditioners of Schur complement and augmented type, for use with minres. By introducing so-called Kronecker product preconditioners we improve the robustness of cheap, mean-based preconditioners with respect to the statistical properties of the stochastically nonlinear diffusion coefficients.

Efficient Iterative Solvers for Stochastic Galerkin Discretizations of Log-Transformed Random Diffusion Problems

“…It is known that A is ill-conditioned with respect to the mesh size h, the standard deviation σ of the logtransformed diffusion coefficient a (M) , and the total degree d of the chaos polynomials; see [17,37,47]. Ill-conditioning with respect to the spatial mesh size can be handled by a mean-based block-diagonal preconditioner derived by approximating the random diffusion coefficient exp(a (M) ) by its mean value exp(a (M) ) , see [23,31,35,36].…”

Solving Log-Transformed Random Diffusion Problems by Stochastic Galerkin Mixed Finite Element Methods

“…If (1.9) holds then the stability of this scheme can be established straight-forwardly (see [6] and [1]). As explained in [14], however, applying SGMFEMs to (1.6)-(1.8) leads to linear systems with block dense and ill-conditioned indefinite matrices that are highly expensive to solve. Our goal is to approximate q efficiently when the diffusion coefficient is the stochastically nonlinear function e a M .…”

“…We further assume that the random variables ξ k : Ω → Γ k ⊂ R are bounded and independent. For simplicity, f is assumed to be a deterministic function of x ∈ D. It is well known (e.g., see [14]) that applying stochastic Galerkin finite element methods (SGFEMs) to PDEs with coefficients of the form e a M results in very large linear systems Ax = b with coefficient matrices of the form…”

Erratum: Preconditioning Stochastic Galerkin Saddle Point Systems

“…Lemma 1 (Lemma 5.3 in [1]). Let X be symmetric and positive definite, and let 0 < ν 1 ≤ · · · ≤ ν n , where n = n ξ n q , be the eigenvalues of X −1 A.…”

“…In the proof given in [1], the two sentences under (5.19) on page 2813 are incorrect. For clarity, we give the entire proof again with the necessary correction.…”