MatExPre: A matrix exponential preconditioner for the high-frequency Helmholtz equation

TL;DR

This paper introduces MatExPre, a novel preconditioner for the high-frequency Helmholtz equation, utilizing matrix exponential properties to improve solver efficiency and scalability in large-scale seismic simulations.

Contribution

The paper develops a new preconditioner based on matrix exponentials, connecting time-domain solvers with exponential integrators for better high-frequency Helmholtz problem solutions.

Findings

Effective in 2D and 3D models

Improves convergence and scalability

Performs well on seismic benchmarks

Abstract

In this article, we present a new preconditioner, MatExPre, for the high-frequency Helmholtz equation by leveraging the properties of matrix exponentials. Our approach begins by reformulating the Helmholtz equation into a Schr\"{o}dinger-like equation and constructing a time-domain solver based on a fixed-point iteration. We then establish a rigorous connection between the time-domain solver and matrix exponential integrators, which enables us to derive algebraic preconditioners that rely solely on sparse matrix-vector products. Spectral analysis and a detailed numerical implementation strategy, including performance improvements achieved through complex shifting, are discussed. Finally, numerical experiments on 2D and large-scale 3D homogeneous and inhomogeneous models, including benchmark seismic examples, substantiate the effectiveness and scalability of the proposed methods.