A low-rank algorithm for solving Lyapunov operator $φ$-functions within the matrix-valued exponential integrators
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866909447809400832 |
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| author | Li, Dongping Zhang, Xiuying Tian, Hongjiong |
| author_facet | Li, Dongping Zhang, Xiuying Tian, Hongjiong |
| contents | In this work we present a low-rank algorithm for computing low-rank approximations of large-scale Lyapunov operator $φ$-functions. These computations play a crucial role in implementing of matrix-valued exponential integrators for large-scale stiff matrix differential equations, where the (approximate) solutions are of low rank.The proposed method employs a scaling and recursive procedure, complemented by a quasi-backward error analysis to determine the optimal parameters. The computational cost is primarily determined by the multiplication of sparse matrices with block vectors. Numerical experiments validate the effectiveness of the proposed method as a foundational tool for matrix-valued exponential integrators in solving differential Lyapunov equations and Riccati equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_02408 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A low-rank algorithm for solving Lyapunov operator $φ$-functions within the matrix-valued exponential integrators Li, Dongping Zhang, Xiuying Tian, Hongjiong Numerical Analysis 65L05, 65F10, 65F30 G.1 In this work we present a low-rank algorithm for computing low-rank approximations of large-scale Lyapunov operator $φ$-functions. These computations play a crucial role in implementing of matrix-valued exponential integrators for large-scale stiff matrix differential equations, where the (approximate) solutions are of low rank.The proposed method employs a scaling and recursive procedure, complemented by a quasi-backward error analysis to determine the optimal parameters. The computational cost is primarily determined by the multiplication of sparse matrices with block vectors. Numerical experiments validate the effectiveness of the proposed method as a foundational tool for matrix-valued exponential integrators in solving differential Lyapunov equations and Riccati equations. |
| title | A low-rank algorithm for solving Lyapunov operator $φ$-functions within the matrix-valued exponential integrators |
| topic | Numerical Analysis 65L05, 65F10, 65F30 G.1 |
| url | https://arxiv.org/abs/2212.02408 |