Block Triangular Preconditioning for Stochastic Galerkin Method

TL;DR

This paper introduces a block triangular preconditioner for iterative solvers applied to large sparse systems from stochastic Galerkin discretizations, improving efficiency especially for high-variance stochastic problems.

Contribution

The paper proposes a novel block triangular preconditioner leveraging the structure of stochastic Galerkin matrices, enhancing solver performance over traditional methods.

Findings

Preconditioner improves convergence speed for high-variance stochastic problems.

Spectral bounds are established for convergence analysis.

Numerical results show superior performance compared to block diagonal preconditioners.

Abstract

In this paper we study fast iterative solvers for the large sparse linear systems resulting from the stochastic Galerkin discretization of stochastic partial differential equations. A block triangular preconditioner is introduced and applied to the Krylov subspace methods, including the generalized minimum residual method and the generalized preconditioned conjugate gradient method. This preconditioner utilizes the special structures of the stochastic Galerkin matrices to achieve high efficiency. Spectral bounds for the preconditioned matrix are provided for convergence analysis. The preconditioner system can be solved approximately by geometric multigrid V-cycle. Numerical results indicate that the block triangular preconditioner has better performance than the traditional block diagonal preconditioner for stochastic problems with large variance.