The Helmholtz equation with uncertainties in the wavenumber

TL;DR

This paper studies the Helmholtz equation with uncertain wavenumber modeled as a random variable or field, discretized with finite differences, and solved using a stochastic Galerkin method with iterative solvers and preconditioners.

Contribution

It introduces a stochastic Galerkin approach for the Helmholtz equation with uncertain wavenumber and proposes effective preconditioners for the resulting high-dimensional linear system.

Findings

Preconditioners significantly reduce iteration steps.

Both preconditioners decrease computation time.

The method effectively handles uncertainty in the wavenumber.

Abstract

We investigate the Helmholtz equation with suitable boundary conditions and uncertainties in the wavenumber. Thus the wavenumber is modeled as a random variable or a random field. We discretize the Helmholtz equation using finite differences in space, which leads to a linear system of algebraic equations including random variables. A stochastic Galerkin method yields a deterministic linear system of algebraic equations. This linear system is high-dimensional, sparse and complex symmetric but, in general, not hermitian. We therefore solve this system iteratively with GMRES and propose two preconditioners: a complex shifted Laplace preconditioner and a mean value preconditioner. Both preconditioners reduce the number of iteration steps as well as the computation time in our numerical experiments.