A Preconditioned Low-Rank Projection Method with a Rank-Reduction Scheme for Stochastic Partial Differential Equations

TL;DR

This paper introduces a novel iterative algorithm with a rank-reduction scheme for efficiently solving large stochastic PDE systems by exploiting tensor structures, significantly reducing computational complexity.

Contribution

It proposes a new low-rank projection method with an on-the-fly rank-reduction scheme tailored for stochastic PDEs, improving computational efficiency.

Findings

Efficiently approximates solutions in low-rank tensor format.

Reduces tensor ranks during iterations to prevent computational blow-up.

Demonstrates effectiveness through numerical experiments on benchmark problems.

Abstract

In this study, we consider the numerical solution of large systems of linear equations obtained from the stochastic Galerkin formulation of stochastic partial differential equations. We propose an iterative algorithm that exploits the Kronecker product structure of the linear systems. The proposed algorithm efficiently approximates the solutions in low-rank tensor format. Using standard Krylov subspace methods for the data in tensor format is computationally prohibitive due to the rapid growth of tensor ranks during the iterations. To keep tensor ranks low over the entire iteration process, we devise a rank-reduction scheme that can be combined with the iterative algorithm. The proposed rank-reduction scheme identifies an important subspace in the stochastic domain and compresses tensors of high rank on-the-fly during the iterations. The proposed reduction scheme is a multilevel method in that the important subspace can be identified inexpensively in a coarse spatial grid setting. The efficiency of the proposed method is illustrated by numerical experiments on benchmark problems.