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arXiv · 2506.03617

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

Shubin Fu·Qing Huo Liu·Qiwei Zhan·Eric T. Chung·Changqing Ye
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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.

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Shubin Fu, Qing Huo Liu, Qiwei Zhan, Eric T. Chung, Changqing Ye. 2025-06-04. MatExPre: A matrix exponential preconditioner for the high-frequency Helmholtz equation. https://arxiv.org/abs/2506.03617

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