Randomized Iterative Solver as Iterative Refinement: A Simple Fix Towards Backward Stability
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912074970431488 |
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| author | Xu, Ruihan Lu, Yiping |
| author_facet | Xu, Ruihan Lu, Yiping |
| contents | Iterative sketching and sketch-and-precondition are well-established randomized algorithms for solving large-scale, over-determined linear least-squares problems. In this paper, we introduce a new perspective that interprets Iterative Sketching and Sketching-and-Precondition as forms of Iterative Refinement. We also examine the numerical stability of two distinct refinement strategies, iterative refinement and recursive refinement, which progressively improve the accuracy of a sketched linear solver. Building on this insight, we propose a novel algorithm, Sketched Iterative and Recursive Refinement (SIRR), which combines both refinement methods. SIRR demonstrates a \emph{four order of magnitude improvement} in backward error compared to iterative sketching, achieved simply by reorganizing the computational order, ensuring that the computed solution exactly solves a modified least-squares system where the coefficient matrix deviates only slightly from the original matrix. To the best of our knowledge, \emph{SIRR is the first asymptotically fast, single-stage randomized least-squares solver that achieves both forward and backward stability}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11115 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Randomized Iterative Solver as Iterative Refinement: A Simple Fix Towards Backward Stability Xu, Ruihan Lu, Yiping Numerical Analysis Computation Iterative sketching and sketch-and-precondition are well-established randomized algorithms for solving large-scale, over-determined linear least-squares problems. In this paper, we introduce a new perspective that interprets Iterative Sketching and Sketching-and-Precondition as forms of Iterative Refinement. We also examine the numerical stability of two distinct refinement strategies, iterative refinement and recursive refinement, which progressively improve the accuracy of a sketched linear solver. Building on this insight, we propose a novel algorithm, Sketched Iterative and Recursive Refinement (SIRR), which combines both refinement methods. SIRR demonstrates a \emph{four order of magnitude improvement} in backward error compared to iterative sketching, achieved simply by reorganizing the computational order, ensuring that the computed solution exactly solves a modified least-squares system where the coefficient matrix deviates only slightly from the original matrix. To the best of our knowledge, \emph{SIRR is the first asymptotically fast, single-stage randomized least-squares solver that achieves both forward and backward stability}. |
| title | Randomized Iterative Solver as Iterative Refinement: A Simple Fix Towards Backward Stability |
| topic | Numerical Analysis Computation |
| url | https://arxiv.org/abs/2410.11115 |