An Algorithm Based on Compute Unified Device Architecture for Estimating Covering Functionals of Convex Bodies
In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a longstanding open problem from Convex and Discrete Geometry, it is essential to estimate covering functionals of convex bodies effectively. Recently, He et al. and Yu et al. provided two deterministic global optimizatio...
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| Format: | Article |
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MDPI AG
2024-02-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/13/2/132 |
| _version_ | 1797298915324723200 |
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| author | Xiangyang Han Senlin Wu Longzhen Zhang |
| author_facet | Xiangyang Han Senlin Wu Longzhen Zhang |
| author_sort | Xiangyang Han |
| collection | DOAJ |
| description | In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a longstanding open problem from Convex and Discrete Geometry, it is essential to estimate covering functionals of convex bodies effectively. Recently, He et al. and Yu et al. provided two deterministic global optimization algorithms having high computational complexity for this purpose. Since satisfactory estimations of covering functionals will be sufficient in Zong’s program, we propose a stochastic global optimization algorithm based on CUDA and provide an error estimation for the algorithm. The accuracy of our algorithm is tested by comparing numerical and exact values of covering functionals of convex bodies including the Euclidean unit disc, the three-dimensional Euclidean unit ball, the regular tetrahedron, and the regular octahedron. We also present estimations of covering functionals for the regular dodecahedron and the regular icosahedron. |
| first_indexed | 2024-03-07T22:42:59Z |
| format | Article |
| id | doaj.art-cbf444e455a241b6ac1c2da03195f0b6 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-07T22:42:59Z |
| publishDate | 2024-02-01 |
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| series | Axioms |
| spelling | doaj.art-cbf444e455a241b6ac1c2da03195f0b62024-02-23T15:07:30ZengMDPI AGAxioms2075-16802024-02-0113213210.3390/axioms13020132An Algorithm Based on Compute Unified Device Architecture for Estimating Covering Functionals of Convex BodiesXiangyang Han0Senlin Wu1Longzhen Zhang2School of Mathematics, North University of China, Taiyuan 030051, ChinaSchool of Mathematics, North University of China, Taiyuan 030051, ChinaSchool of Mathematics, North University of China, Taiyuan 030051, ChinaIn Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a longstanding open problem from Convex and Discrete Geometry, it is essential to estimate covering functionals of convex bodies effectively. Recently, He et al. and Yu et al. provided two deterministic global optimization algorithms having high computational complexity for this purpose. Since satisfactory estimations of covering functionals will be sufficient in Zong’s program, we propose a stochastic global optimization algorithm based on CUDA and provide an error estimation for the algorithm. The accuracy of our algorithm is tested by comparing numerical and exact values of covering functionals of convex bodies including the Euclidean unit disc, the three-dimensional Euclidean unit ball, the regular tetrahedron, and the regular octahedron. We also present estimations of covering functionals for the regular dodecahedron and the regular icosahedron.https://www.mdpi.com/2075-1680/13/2/132Hadwiger’s covering problemcovering functionalsCUDAerror estimation |
| spellingShingle | Xiangyang Han Senlin Wu Longzhen Zhang An Algorithm Based on Compute Unified Device Architecture for Estimating Covering Functionals of Convex Bodies Axioms Hadwiger’s covering problem covering functionals CUDA error estimation |
| title | An Algorithm Based on Compute Unified Device Architecture for Estimating Covering Functionals of Convex Bodies |
| title_full | An Algorithm Based on Compute Unified Device Architecture for Estimating Covering Functionals of Convex Bodies |
| title_fullStr | An Algorithm Based on Compute Unified Device Architecture for Estimating Covering Functionals of Convex Bodies |
| title_full_unstemmed | An Algorithm Based on Compute Unified Device Architecture for Estimating Covering Functionals of Convex Bodies |
| title_short | An Algorithm Based on Compute Unified Device Architecture for Estimating Covering Functionals of Convex Bodies |
| title_sort | algorithm based on compute unified device architecture for estimating covering functionals of convex bodies |
| topic | Hadwiger’s covering problem covering functionals CUDA error estimation |
| url | https://www.mdpi.com/2075-1680/13/2/132 |
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