A Low-Rank Multigrid Method for the Stochastic Steady-State Diffusion Problem

TL;DR

This paper introduces a low-rank multigrid method tailored for efficiently solving large stochastic PDE systems with tensor structures, significantly reducing computational costs while maintaining convergence.

Contribution

It proposes a novel low-rank multigrid algorithm with proven convergence for stochastic PDEs, improving efficiency over traditional methods.

Findings

Effective in reducing memory and computational effort.

Proven convergence with analytic error bounds.

Outperforms standard multigrid in large-scale problems.

Abstract

We study a multigrid method for solving large linear systems of equations with tensor product structure. Such systems are obtained from stochastic finite element discretization of stochastic partial differential equations such as the steady-state diffusion problem with random coefficients. When the variance in the problem is not too large, the solution can be well approximated by a low-rank object. In the proposed multigrid algorithm, the matrix iterates are truncated to low rank to reduce memory requirements and computational effort. The method is proved convergent with an analytic error bound. Numerical experiments show its effectiveness in solving the Galerkin systems compared to the original multigrid solver, especially when the number of degrees of freedom associated with the spatial discretization is large.