A CUR Krylov Solver for Large-Scale Linear Matrix Equations
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911271754924032 |
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| author | Akbari, Saeed Lombardi, Damiano Babaee, Hessam |
| author_facet | Akbari, Saeed Lombardi, Damiano Babaee, Hessam |
| contents | Developing efficient solvers for large-scale multi-term linear matrix equations remains a central challenge in numerical linear algebra and is still largely unresolved. This paper introduces a methodology leveraging CUR decomposition for solving large-scale generalized Sylvester as well as non-Sylvester multi-term equations on low-rank matrix manifolds. The approach decomposes the original equation into two smaller subproblems: one involving all columns with a small subset of rows, and the other involving all rows with a small subset of columns. The rows and columns are strategically selected using the discrete empirical interpolation method. We further utilize the CUR properties and propose a novel iterative scheme that removes the dependencies between selected and unselected rows (and likewise for columns), thereby enabling the subset problems to be solved independently. We present a Krylov-based scheme for solving the resulting subproblems, which scales effectively to large problems and does not rely on a Sylvester structure. The method incorporates rank adaptivity, dynamically adjusting computational rank to reach the desired accuracy. The methodology is demonstrated in three representative settings: (i) implicit time integration of matrix differential equations on low-rank manifolds, leading to multi-term linear matrix equations; (ii) large-scale steady-state generalized Lyapunov equations including cases of size up to $10^{13}$ unknown entries; and (iii) non-Sylvester linear matrix equations with Hadamard product terms, such as those arising in nonlinear partial differential equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_14015 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A CUR Krylov Solver for Large-Scale Linear Matrix Equations Akbari, Saeed Lombardi, Damiano Babaee, Hessam Numerical Analysis Developing efficient solvers for large-scale multi-term linear matrix equations remains a central challenge in numerical linear algebra and is still largely unresolved. This paper introduces a methodology leveraging CUR decomposition for solving large-scale generalized Sylvester as well as non-Sylvester multi-term equations on low-rank matrix manifolds. The approach decomposes the original equation into two smaller subproblems: one involving all columns with a small subset of rows, and the other involving all rows with a small subset of columns. The rows and columns are strategically selected using the discrete empirical interpolation method. We further utilize the CUR properties and propose a novel iterative scheme that removes the dependencies between selected and unselected rows (and likewise for columns), thereby enabling the subset problems to be solved independently. We present a Krylov-based scheme for solving the resulting subproblems, which scales effectively to large problems and does not rely on a Sylvester structure. The method incorporates rank adaptivity, dynamically adjusting computational rank to reach the desired accuracy. The methodology is demonstrated in three representative settings: (i) implicit time integration of matrix differential equations on low-rank manifolds, leading to multi-term linear matrix equations; (ii) large-scale steady-state generalized Lyapunov equations including cases of size up to $10^{13}$ unknown entries; and (iii) non-Sylvester linear matrix equations with Hadamard product terms, such as those arising in nonlinear partial differential equations. |
| title | A CUR Krylov Solver for Large-Scale Linear Matrix Equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2511.14015 |