Low-rank solutions to the stochastic Helmholtz equation

TL;DR

This paper develops low-rank approximation methods for the stochastic Helmholtz equation, demonstrating efficiency in computation and storage, and introduces a low-rank preconditioning approach.

Contribution

It establishes the existence of low-rank solutions for the stochastic Helmholtz equation and proposes an efficient low-rank algorithm with a novel preconditioning strategy.

Findings

Low-rank solutions reduce computational cost and storage.

Numerical results confirm efficiency gains over full-rank methods.

A new low-rank preconditioner improves iterative solver performance.

Abstract

In this paper, we consider low-rank approximations for the solutions to the stochastic Helmholtz equation with random coefficients. A Stochastic Galerkin finite element method is used for the discretization of the Helmholtz problem. Existence theory for the low-rank approximation is established when the system matrix is indefinite. The low-rank algorithm does not require the construction of a large system matrix which results in an advantage in terms of CPU time and storage. Numerical results show that, when the operations in a low-rank method are performed efficiently, it is possible to obtain an advantage in terms of storage and CPU time compared to computations in full rank. We also propose a general approach to implement a preconditioner using the low-rank format efficiently.