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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.15322 |
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| _version_ | 1866908457862430720 |
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| author | Ling, Yonghui Xiong, Zikang Liang, Juan |
| author_facet | Ling, Yonghui Xiong, Zikang Liang, Juan |
| contents | This work investigates the local convergence behavior of Anderson acceleration in solving nonlinear systems. We establish local R-linear convergence results for Anderson acceleration with general depth $m$ under the assumptions that the Jacobian of the nonlinear operator is Hölder continuous and the corresponding fixed-point function is contractive. In the Lipschitz continuous case, we obtain a sharper R-linear convergence factor. We also derive a refined residual bound for the depth $m = 1$ under the same assumptions used for the general depth results. Applications to a nonsymmetric Riccati equation from transport theory demonstrate that Anderson acceleration yields comparable results to several existing fixed-point methods for the regular cases, and that it brings significant reductions in both the number of iterations and computation time, even in challenging cases involving nearly singular or large-scale problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15322 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence analysis of Anderson acceleration for nonlinear equations with Hölder continuous derivatives Ling, Yonghui Xiong, Zikang Liang, Juan Numerical Analysis 65H10, 15A24 This work investigates the local convergence behavior of Anderson acceleration in solving nonlinear systems. We establish local R-linear convergence results for Anderson acceleration with general depth $m$ under the assumptions that the Jacobian of the nonlinear operator is Hölder continuous and the corresponding fixed-point function is contractive. In the Lipschitz continuous case, we obtain a sharper R-linear convergence factor. We also derive a refined residual bound for the depth $m = 1$ under the same assumptions used for the general depth results. Applications to a nonsymmetric Riccati equation from transport theory demonstrate that Anderson acceleration yields comparable results to several existing fixed-point methods for the regular cases, and that it brings significant reductions in both the number of iterations and computation time, even in challenging cases involving nearly singular or large-scale problems. |
| title | Convergence analysis of Anderson acceleration for nonlinear equations with Hölder continuous derivatives |
| topic | Numerical Analysis 65H10, 15A24 |
| url | https://arxiv.org/abs/2507.15322 |