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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2405.16040 |
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| _version_ | 1866929358107574272 |
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| author | Hu, Shilong Liu, Hao Wang, Dong |
| author_facet | Hu, Shilong Liu, Hao Wang, Dong |
| contents | In this paper, we introduce two iterative methods for longest minimal length partition problem, which asks whether the disc (ball) is the set maximizing the total perimeter of the shortest partition that divides the total region into sub-regions with given volume proportions, under a volume constraint. The objective functional is approximated by a short-time heat flow using indicator functions of regions and Gaussian convolution. The problem is then represented as a constrained max-min optimization problem. Auction dynamics is used to find the shortest partition in a fixed region, and threshold dynamics is used to update the region. Numerical experiments in two-dimensional and three-dimensional cases are shown with different numbers of partitions, unequal volume proportions, and different initial shapes. The results of both methods are consistent with the conjecture that the disc in two dimensions and the ball in three dimensions are the solution of the longest minimal length partition problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16040 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Iterative Thresholding Methods for Longest Minimal Length Partitions Hu, Shilong Liu, Hao Wang, Dong Numerical Analysis 49M20, 49Q05, 49M05, 41A30 In this paper, we introduce two iterative methods for longest minimal length partition problem, which asks whether the disc (ball) is the set maximizing the total perimeter of the shortest partition that divides the total region into sub-regions with given volume proportions, under a volume constraint. The objective functional is approximated by a short-time heat flow using indicator functions of regions and Gaussian convolution. The problem is then represented as a constrained max-min optimization problem. Auction dynamics is used to find the shortest partition in a fixed region, and threshold dynamics is used to update the region. Numerical experiments in two-dimensional and three-dimensional cases are shown with different numbers of partitions, unequal volume proportions, and different initial shapes. The results of both methods are consistent with the conjecture that the disc in two dimensions and the ball in three dimensions are the solution of the longest minimal length partition problem. |
| title | Iterative Thresholding Methods for Longest Minimal Length Partitions |
| topic | Numerical Analysis 49M20, 49Q05, 49M05, 41A30 |
| url | https://arxiv.org/abs/2405.16040 |