Factorizing the Stochastic Galerkin System

TL;DR

This paper introduces a factorization technique for the matrix systems from stochastic Galerkin methods, providing bounds, preconditioning insights, and flexible implementation for PDEs with stochastic inputs.

Contribution

It derives a novel factorization of the parameter-dependent matrix system, offering eigenvalue bounds, preconditioning guidance, and adaptable implementation strategies.

Findings

Eigenvalue bounds for the factorized system

Effective preconditioners demonstrated on elliptic PDEs

Successful application to CFD problems

Abstract

Recent work has explored solver strategies for the linear system of equations arising from a spectral Galerkin approximation of the solution of PDEs with parameterized (or stochastic) inputs. We consider the related problem of a matrix equation whose matrix and right hand side depend on a set of parameters (e.g. a PDE with stochastic inputs semidiscretized in space) and examine the linear system arising from a similar Galerkin approximation of the solution. We derive a useful factorization of this system of equations, which yields bounds on the eigenvalues, clues to preconditioning, and a flexible implementation method for a wide array of problems. We complement this analysis with (i) a numerical study of preconditioners on a standard elliptic PDE test problem and (ii) a fluids application using existing CFD codes; the MATLAB codes used in the numerical studies are available online.