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Multipreconditioning with directional sweeping methods for high-frequency Helmholtz problems

Niall Bootland, Tyrone Rees

TL;DR

This paper addresses solving high-frequency Helmholtz problems by using multipreconditioning with directional sweeping preconditioners within MPGMRES. The approach combines simple, directionally sweeping subdomain solves with approximate transparent boundary conditions as multiple preconditioners, enabling parallel application within the Krylov solver. Key findings show that combining orthogonal sweeps (e.g., left-to-right-left with bottom-to-top-bottom) significantly reduces iterations compared to a single sweep, with robustness up to high $k$ when the mesh resolves about 10 points per wavelength. The work demonstrates potential for near-linear scaling in practice, provides insights into preconditioner design, and outlines avenues for extension to higher-order transmission conditions, 3D problems, and Maxwell equations.

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.

Multipreconditioning with directional sweeping methods for high-frequency Helmholtz problems

TL;DR

This paper addresses solving high-frequency Helmholtz problems by using multipreconditioning with directional sweeping preconditioners within MPGMRES. The approach combines simple, directionally sweeping subdomain solves with approximate transparent boundary conditions as multiple preconditioners, enabling parallel application within the Krylov solver. Key findings show that combining orthogonal sweeps (e.g., left-to-right-left with bottom-to-top-bottom) significantly reduces iterations compared to a single sweep, with robustness up to high when the mesh resolves about 10 points per wavelength. The work demonstrates potential for near-linear scaling in practice, provides insights into preconditioner design, and outlines avenues for extension to higher-order transmission conditions, 3D problems, and Maxwell equations.

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.
Paper Structure (4 sections, 6 equations, 1 figure, 5 tables)

This paper contains 4 sections, 6 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Sweeping Methods For Helmholtz Problems
  3. Multipreconditioned GMRES
  4. Numerical Results

Figures (1)

  • Figure 1: Sequential decomposition into $N$ subdomains $\Omega_{s}$, $1 \le s \le N$, with the internal subdomain boundaries $\Gamma_{s,1}$ (for $2 \le s \le N$) and $\Gamma_{s,2}$ (for $1 \le s \le N-1$) ordered sequentially as with the decomposition, here left-to-right in this illustration; the overlap regions are shaded darker.