Truncated hierarchical preconditioning for the stochastic Galerkin FEM

TL;DR

This paper introduces two novel preconditioning strategies for stochastic Galerkin FEM systems, leveraging hierarchical matrix structures and decay properties to improve iterative solver efficiency.

Contribution

The authors develop and analyze two new preconditioners that exploit hierarchical structures and decay in stochastic Galerkin matrices, reducing computational cost.

Findings

Preconditioners improve convergence of Krylov solvers.

Decay properties enable matrix-vector multiplication truncation.

Numerical experiments demonstrate effectiveness.

Abstract

Stochastic Galerkin finite element discretizations of partial differential equations with coefficients characterized by arbitrary distributions lead, in general, to fully block dense linear systems. We propose two novel strategies for constructing preconditioners for these systems to be used with Krylov subspace iterative solvers. In particular, we present a variation on of the hierarchical Schur complement preconditioner, developed recently by the authors, and an adaptation of the symmetric block Gauss-Seidel method. Both preconditioners take advantage of the hierarchical structure of global stochastic Galerkin matrices, and also, when applicable, of the decay of the norms of the stiffness matrices obtained from the polynomial chaos expansion of the coefficients. This decay allows to truncate the matrix-vector multiplications in the action of the preconditioners. Also, throughout the global matrix hierarchy, we approximate solves with certain submatrices by the associated diagonal block solves. The preconditioners thus require only a limited number of stiffness matrices obtained from the polynomial chaos expansion of the coefficients, and a preconditioner for the diagonal blocks of the global matrix. The performance is illustrated by numerical experiments.