MatExPre: A matrix exponential preconditioner for the high-frequency Helmholtz equation
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866913874938167296 |
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| author | Fu, Shubin Liu, Qing Huo Zhan, Qiwei Chung, Eric T. Ye, Changqing |
| author_facet | Fu, Shubin Liu, Qing Huo Zhan, Qiwei Chung, Eric T. Ye, Changqing |
| contents | 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 Schrödinger-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_03617 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | MatExPre: A matrix exponential preconditioner for the high-frequency Helmholtz equation Fu, Shubin Liu, Qing Huo Zhan, Qiwei Chung, Eric T. Ye, Changqing Numerical Analysis 65M22, 65F08 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 Schrödinger-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. |
| title | MatExPre: A matrix exponential preconditioner for the high-frequency Helmholtz equation |
| topic | Numerical Analysis 65M22, 65F08 |
| url | https://arxiv.org/abs/2506.03617 |