lrAA: Low-Rank Anderson Acceleration
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913751035281408 |
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| author | Appelo, Daniel Cheng, Yingda |
| author_facet | Appelo, Daniel Cheng, Yingda |
| contents | This paper proposes a new framework for computing low-rank solutions to nonlinear matrix equations arising from spatial discretization of nonlinear partial differential equations: low-rank Anderson acceleration (lrAA). lrAA is an adaptation of Anderson acceleration (AA), a well-known approach for solving nonlinear fixed point problems, to the low-rank format. In particular, lrAA carries out all linear and nonlinear operations in low-rank form with rank truncation using an adaptive truncation tolerance. We propose a simple scheduling strategy to update the truncation tolerance throughout the iteration according to a residual indicator. This controls the intermediate rank and iteration number effectively. To perform rank truncation for nonlinear functions, we propose a new cross approximation, which we call Cross-DEIM, with adaptive error control that is based on the discrete empirical interpolation method (DEIM). Cross-DEIM employs an iterative update between the approximate singular value decomposition (SVD) and cross approximation. It naturally incorporates a warm-start strategy for each lrAA iterate. We demonstrate the superior performance of lrAA applied to a range of linear and nonlinear problems, including those arising from finite difference discretizations of Laplace's equation, the Bratu problem, the elliptic Monge-Ampére equation and the Allen-Cahn equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_03909 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | lrAA: Low-Rank Anderson Acceleration Appelo, Daniel Cheng, Yingda Numerical Analysis This paper proposes a new framework for computing low-rank solutions to nonlinear matrix equations arising from spatial discretization of nonlinear partial differential equations: low-rank Anderson acceleration (lrAA). lrAA is an adaptation of Anderson acceleration (AA), a well-known approach for solving nonlinear fixed point problems, to the low-rank format. In particular, lrAA carries out all linear and nonlinear operations in low-rank form with rank truncation using an adaptive truncation tolerance. We propose a simple scheduling strategy to update the truncation tolerance throughout the iteration according to a residual indicator. This controls the intermediate rank and iteration number effectively. To perform rank truncation for nonlinear functions, we propose a new cross approximation, which we call Cross-DEIM, with adaptive error control that is based on the discrete empirical interpolation method (DEIM). Cross-DEIM employs an iterative update between the approximate singular value decomposition (SVD) and cross approximation. It naturally incorporates a warm-start strategy for each lrAA iterate. We demonstrate the superior performance of lrAA applied to a range of linear and nonlinear problems, including those arising from finite difference discretizations of Laplace's equation, the Bratu problem, the elliptic Monge-Ampére equation and the Allen-Cahn equation. |
| title | lrAA: Low-Rank Anderson Acceleration |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2503.03909 |