Anderson Acceleration with Truncated Gram-Schmidt
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929421681688576 |
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| author | Tang, Ziyuan Xu, Tianshi He, Huan Saad, Yousef Xi, Yuanzhe |
| author_facet | Tang, Ziyuan Xu, Tianshi He, Huan Saad, Yousef Xi, Yuanzhe |
| contents | Anderson Acceleration (AA) is a popular algorithm designed to enhance the convergence of fixed-point iterations. In this paper, we introduce a variant of AA based on a Truncated Gram-Schmidt process (AATGS) which has a few advantages over the classical AA. In particular, an attractive feature of AATGS is that its iterates obey a three-term recurrence in the situation when it is applied to solving symmetric linear problems and this can lead to a considerable reduction of memory and computational costs. We analyze the convergence of AATGS in both full-depth and limited-depth scenarios and establish its equivalence to the classical AA in the linear case. We also report on the effectiveness of AATGS through a set of numerical experiments, ranging from solving nonlinear partial differential equations to tackling nonlinear optimization problems. In particular, the performance of the method is compared with that of the classical AA algorithms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_14961 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Anderson Acceleration with Truncated Gram-Schmidt Tang, Ziyuan Xu, Tianshi He, Huan Saad, Yousef Xi, Yuanzhe Numerical Analysis 65F10, 68W25, 65B99, 65N22 Anderson Acceleration (AA) is a popular algorithm designed to enhance the convergence of fixed-point iterations. In this paper, we introduce a variant of AA based on a Truncated Gram-Schmidt process (AATGS) which has a few advantages over the classical AA. In particular, an attractive feature of AATGS is that its iterates obey a three-term recurrence in the situation when it is applied to solving symmetric linear problems and this can lead to a considerable reduction of memory and computational costs. We analyze the convergence of AATGS in both full-depth and limited-depth scenarios and establish its equivalence to the classical AA in the linear case. We also report on the effectiveness of AATGS through a set of numerical experiments, ranging from solving nonlinear partial differential equations to tackling nonlinear optimization problems. In particular, the performance of the method is compared with that of the classical AA algorithms. |
| title | Anderson Acceleration with Truncated Gram-Schmidt |
| topic | Numerical Analysis 65F10, 68W25, 65B99, 65N22 |
| url | https://arxiv.org/abs/2403.14961 |