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An Algorithm Based on CUDA for Estimating Covering Functionals of Convex Bodies
Version 1
: Received: 16 January 2024 / Approved: 16 January 2024 / Online: 16 January 2024 (12:32:18 CET)
A peer-reviewed article of this Preprint also exists.
Han, X.; Wu, S.; Zhang, L. An Algorithm Based on Compute Unified Device Architecture for Estimating Covering Functionals of Convex Bodies. Axioms 2024, 13, 132. Han, X.; Wu, S.; Zhang, L. An Algorithm Based on Compute Unified Device Architecture for Estimating Covering Functionals of Convex Bodies. Axioms 2024, 13, 132.
Abstract
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. [J. Optim. Theory Appl. 2023, 196, 1036–1055.] and Yu et al. [Mathematics, 2023, 11, 2000] 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.
Keywords
Hadwiger’s covering problem; covering functionals; CUDA; error estimation
Subject
Computer Science and Mathematics, Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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