Splitting-based randomized dynamical low-rank approximations for stiff matrix differential equations
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915547891892224 |
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| author | Wu, Zi Zhao, Yong-Liang Gu, Xian-Ming |
| author_facet | Wu, Zi Zhao, Yong-Liang Gu, Xian-Ming |
| contents | In the fields of control theory and machine learning, the dynamic low-rank approximation for large-scale matrices has received substantial attention. Considering large-scale semilinear stiff matrix differential equations, we propose splitting-based randomized dynamical low-rank approximations for a low-rank solution of the stiff matrix differential equation. We first split such the equation into a stiff linear subproblem and a nonstiff nonlinear subproblem. Then, a low-rank exponential integrator is applied to the linear subproblem. Two randomized low-rank approaches are employed for the nonlinear subproblem. Furthermore, we extend the proposed methods to rank-adaptation scenarios. Through rigorous validation on canonical stiff matrix differential problems, including spatially discretized Allen-Cahn equations and differential Riccati equations, we demonstrate that our methods achieve desired convergence orders. Numerical results confirm the robustness and accuracy of the proposed methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_15259 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Splitting-based randomized dynamical low-rank approximations for stiff matrix differential equations Wu, Zi Zhao, Yong-Liang Gu, Xian-Ming Numerical Analysis In the fields of control theory and machine learning, the dynamic low-rank approximation for large-scale matrices has received substantial attention. Considering large-scale semilinear stiff matrix differential equations, we propose splitting-based randomized dynamical low-rank approximations for a low-rank solution of the stiff matrix differential equation. We first split such the equation into a stiff linear subproblem and a nonstiff nonlinear subproblem. Then, a low-rank exponential integrator is applied to the linear subproblem. Two randomized low-rank approaches are employed for the nonlinear subproblem. Furthermore, we extend the proposed methods to rank-adaptation scenarios. Through rigorous validation on canonical stiff matrix differential problems, including spatially discretized Allen-Cahn equations and differential Riccati equations, we demonstrate that our methods achieve desired convergence orders. Numerical results confirm the robustness and accuracy of the proposed methods. |
| title | Splitting-based randomized dynamical low-rank approximations for stiff matrix differential equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2506.15259 |