Rejuvenating AMLI-Cycle: From Chebyshev Polynomials to Momentum Acceleration
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arXiv
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| Natura: | Preprint |
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2020
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| _version_ | 1866909648652599296 |
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| author | Niu, Chunyan He, Yunhui Hu, Xiaozhe |
| author_facet | Niu, Chunyan He, Yunhui Hu, Xiaozhe |
| contents | In this paper, we investigate the AMLI-cycle method and make two contributions. First, we revisit the AMLI-cycle using the Chebyshev polynomials and establish a theory for its uniform convergence, assuming the two-grid method converges uniformly. This removes the need for estimating extreme eigenvalues at all coarse levels. Only an estimation of the two-grid convergence rate is needed, which could be done on the second coarsest level, simplifying implementation and reducing computational costs for large-scale problems. Second, we introduce a momentum-accelerated AMLI-cycle using polynomials from momentum accelerations. This novel approach ensures a uniform condition number without requiring extreme eigenvalue or two-grid convergence rate estimations, making its implementation as straightforward as standard multigrid methods. We prove that it is asymptotically as good as the AMLI-cycle using the Chebyshev polynomials when the quadratic momentum-accelerated polynomials is used. Numerical experiments confirm the robustness and efficiency of the momentum-accelerated AMLI-cycle across various problems, demonstrating performance comparable to the Chebyshev-based AMLI-cycle. These findings validate the theoretical advantages and practical efficacy of the momentum-accelerated AMLI-cycle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2006_16986 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Rejuvenating AMLI-Cycle: From Chebyshev Polynomials to Momentum Acceleration Niu, Chunyan He, Yunhui Hu, Xiaozhe Numerical Analysis 65N55, 65F08, 65F10, 65B99 In this paper, we investigate the AMLI-cycle method and make two contributions. First, we revisit the AMLI-cycle using the Chebyshev polynomials and establish a theory for its uniform convergence, assuming the two-grid method converges uniformly. This removes the need for estimating extreme eigenvalues at all coarse levels. Only an estimation of the two-grid convergence rate is needed, which could be done on the second coarsest level, simplifying implementation and reducing computational costs for large-scale problems. Second, we introduce a momentum-accelerated AMLI-cycle using polynomials from momentum accelerations. This novel approach ensures a uniform condition number without requiring extreme eigenvalue or two-grid convergence rate estimations, making its implementation as straightforward as standard multigrid methods. We prove that it is asymptotically as good as the AMLI-cycle using the Chebyshev polynomials when the quadratic momentum-accelerated polynomials is used. Numerical experiments confirm the robustness and efficiency of the momentum-accelerated AMLI-cycle across various problems, demonstrating performance comparable to the Chebyshev-based AMLI-cycle. These findings validate the theoretical advantages and practical efficacy of the momentum-accelerated AMLI-cycle. |
| title | Rejuvenating AMLI-Cycle: From Chebyshev Polynomials to Momentum Acceleration |
| topic | Numerical Analysis 65N55, 65F08, 65F10, 65B99 |
| url | https://arxiv.org/abs/2006.16986 |