Multipreconditioning with directional sweeping methods for high-frequency Helmholtz problems

TL;DR

This paper explores a novel multipreconditioning approach using directional sweeping methods to efficiently solve large, complex high-frequency Helmholtz problems, demonstrating promising initial numerical results.

Contribution

It introduces a simple, parallelizable multipreconditioning technique based on directional sweeping methods for high-frequency Helmholtz problems, offering an alternative to more complex existing methods.

Findings

Preliminary results show potential for the proposed approach.

The method is suitable for parallel implementation.

Initial benchmarks indicate improved efficiency.

Abstract

We consider the use of multipreconditioning, which allows for multiple preconditioners to be applied in parallel, on high-frequency Helmholtz problems. Typical applications present challenging sparse linear systems which are complex non-Hermitian and, due to the pollution effect, either very large or else still large but under-resolved in terms of the physics. These factors make finding general purpose, efficient and scalable solvers difficult and no one approach has become the clear method of choice. In this work we take inspiration from domain decomposition strategies known as sweeping methods, which have gained notable interest for their ability to yield nearly-linear asymptotic complexity and which can also be favourable for high-frequency problems. While successful approaches exist, such as those based on higher-order interface conditions, perfectly matched layers (PMLs), or complex tracking of wave fronts, they can often be quite involved or tedious to implement. We investigate here the use of simple sweeping techniques applied in different directions which can then be incorporated in parallel into a multipreconditioned GMRES strategy. Preliminary numerical results on a two-dimensional benchmark problem will demonstrate the potential of this approach.