A Convergent Inexact Abedin-Kitagawa Iteration Method for Monge-Ampère Eigenvalue Problems
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866917085132619776 |
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| author | Chen, Liang Lin, Youyicun Yang, Junqi Yi, Wenfan |
| author_facet | Chen, Liang Lin, Youyicun Yang, Junqi Yi, Wenfan |
| contents | This paper proposes an inexact Aleksandrov-solution-based iteration method, formulated by adapting the convergent Rayleigh inverse iterative scheme introduced by Abedin and Kitagawa, to solve real Monge-Amp{è}re eigenvalue (MAE) problems. The central feature of the proposed approach is the introduction of a flexible error tolerance criterion for computing inexact Aleksandrov solutions to the required subproblems. This allows the inner iteration to be solved approximately without compromising the global convergence properties of the overall scheme, as we established under a ${\cal C}^{2,α}$ boundary condition, and has the potential of achieving reduced computational cost compared to the original algorithm. In practice, for both two- and three-dimensional problems, by leveraging the flexibility of the inexact iterative formulation in conjunction with a fixed-point approach for solving the subproblems, the proposed method performs several times faster than its original version of Abedin and Kitagawa, across all tested problem instances in the numerical experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06160 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Convergent Inexact Abedin-Kitagawa Iteration Method for Monge-Ampère Eigenvalue Problems Chen, Liang Lin, Youyicun Yang, Junqi Yi, Wenfan Numerical Analysis 65H17, 35J96, 65N25 This paper proposes an inexact Aleksandrov-solution-based iteration method, formulated by adapting the convergent Rayleigh inverse iterative scheme introduced by Abedin and Kitagawa, to solve real Monge-Amp{è}re eigenvalue (MAE) problems. The central feature of the proposed approach is the introduction of a flexible error tolerance criterion for computing inexact Aleksandrov solutions to the required subproblems. This allows the inner iteration to be solved approximately without compromising the global convergence properties of the overall scheme, as we established under a ${\cal C}^{2,α}$ boundary condition, and has the potential of achieving reduced computational cost compared to the original algorithm. In practice, for both two- and three-dimensional problems, by leveraging the flexibility of the inexact iterative formulation in conjunction with a fixed-point approach for solving the subproblems, the proposed method performs several times faster than its original version of Abedin and Kitagawa, across all tested problem instances in the numerical experiments. |
| title | A Convergent Inexact Abedin-Kitagawa Iteration Method for Monge-Ampère Eigenvalue Problems |
| topic | Numerical Analysis 65H17, 35J96, 65N25 |
| url | https://arxiv.org/abs/2505.06160 |