Low-synchronization Arnoldi Methods for the Matrix Exponential with Application to Exponential Integrators
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| Format: | Preprint |
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2024
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| author | Tafolla, Tanya Gaudreault, Stéphane Tokman, Mayya |
| author_facet | Tafolla, Tanya Gaudreault, Stéphane Tokman, Mayya |
| contents | High order exponential integrators require computing linear combination of exponential like $φ$-functions of large matrices $A$ times a vector $v$. Krylov projection methods are the most general and remain an efficient choice for computing the matrix-function-vector-product evaluation when the matrix is $A$ is large and unable to be explicitly stored, or when obtaining information about the spectrum is expensive. The Krylov approximation relies on the Gram-Schmidt (GS) orthogonalization procedure to produce the orthonormal basis $V_m$. In parallel, GS orthogonalization requires \textit{global synchronizations} for inner products and vector normalization in the orthogonalization process. Reducing the amount of global synchronizations is of paramount importance for the efficiency of a numerical algorithm in a massively parallel setting. We improve the parallel strong scaling properties of exponential integrators by addressing the underlying bottleneck in the linear algebra using low-synchronization GS methods. The resulting orthogonalization algorithms have an accuracy comparable to modified Gram-Schmidt yet are better suited for distributed architecture, as only one global communication is required per orthogonalization-step. We present geophysics-based numerical experiments and standard examples routinely used to test stiff time integrators, which validate that reducing global communication leads to better parallel scalability and reduced time-to-solution for exponential integrators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_14917 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Low-synchronization Arnoldi Methods for the Matrix Exponential with Application to Exponential Integrators Tafolla, Tanya Gaudreault, Stéphane Tokman, Mayya Numerical Analysis 65F60, 65F25 High order exponential integrators require computing linear combination of exponential like $φ$-functions of large matrices $A$ times a vector $v$. Krylov projection methods are the most general and remain an efficient choice for computing the matrix-function-vector-product evaluation when the matrix is $A$ is large and unable to be explicitly stored, or when obtaining information about the spectrum is expensive. The Krylov approximation relies on the Gram-Schmidt (GS) orthogonalization procedure to produce the orthonormal basis $V_m$. In parallel, GS orthogonalization requires \textit{global synchronizations} for inner products and vector normalization in the orthogonalization process. Reducing the amount of global synchronizations is of paramount importance for the efficiency of a numerical algorithm in a massively parallel setting. We improve the parallel strong scaling properties of exponential integrators by addressing the underlying bottleneck in the linear algebra using low-synchronization GS methods. The resulting orthogonalization algorithms have an accuracy comparable to modified Gram-Schmidt yet are better suited for distributed architecture, as only one global communication is required per orthogonalization-step. We present geophysics-based numerical experiments and standard examples routinely used to test stiff time integrators, which validate that reducing global communication leads to better parallel scalability and reduced time-to-solution for exponential integrators. |
| title | Low-synchronization Arnoldi Methods for the Matrix Exponential with Application to Exponential Integrators |
| topic | Numerical Analysis 65F60, 65F25 |
| url | https://arxiv.org/abs/2410.14917 |