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 Zh. Vychisl. Mat. Mat. Fiz., 2013, Volume 53, Number 2, Pages 181–194 (Mi zvmmf9775)

Iterative method for constructing coverings of the multidimensional unit sphere

G. K. Kameneva, A. V. Lotova, T. S. Mayskayab

a Dorodnitsyn Computing Centre of the Russian Academy of Sciences, Moscow
b M. V. Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics

Abstract: The stepwise-supplement-of-a-covering (SSC) method is described and examined. The method is intended for the numerical construction of near optimal coverings of the multidimensional unit sphere by neighborhoods of a finite number of points (covering basis). Coverings of the unit sphere are used, for example, in nonadaptive polyhedral approximation of multidimensional convex compact bodies based on the evaluation of their support function for directions defined by points of the covering basis. The SSC method is used to iteratively construct a sequence of coverings, each differing from the previous one by a single new point included in the covering basis. Although such coverings are not optimal, it is theoretically shown that they are asymptotically suboptimal. By applying an experimental analysis, the asymptotic efficiency of the SSC method is estimated and the method is shown to be relatively efficient for a small number of points in the covering basis.

Key words: methods for covering the multidimensional unit sphere, interactive method, stepwise-supplement-of-a-covering method, asymptotically suboptimal covering.

DOI: https://doi.org/10.7868/S0044466913020117

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English version:
Computational Mathematics and Mathematical Physics, 2013, 53:2, 131–143

Bibliographic databases:

UDC: 519.6

Citation: G. K. Kamenev, A. V. Lotov, T. S. Mayskaya, “Iterative method for constructing coverings of the multidimensional unit sphere”, Zh. Vychisl. Mat. Mat. Fiz., 53:2 (2013), 181–194; Comput. Math. Math. Phys., 53:2 (2013), 131–143

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. G. K. Kamenev, “Method for polyhedral approximation of a ball with an optimal order of growth of the facet structure cardinality”, Comput. Math. Math. Phys., 54:8 (2014), 1201–1213
2. V. E. Berezkin, A. V. Lotov, “Comparison of two Pareto frontier approximations”, Comput. Math. Math. Phys., 54:9 (2014), 1402–1410
3. G. K. Kamenev, “Asymptotic properties of the estimate refinement method in polyhedral approximation of multidimensional balls”, Comput. Math. Math. Phys., 55:10 (2015), 1619–1632
4. G. K. Kamenev, “Efficiency of the estimate refinement method for polyhedral approximation of multidimensional balls”, Comput. Math. Math. Phys., 56:5 (2016), 744–755
5. A. V. Lotov, “Method for constructing an external polyhedral estimate of the trajectory tube for a nonlinear dynamic system”, Dokl. Math., 95:1 (2017), 95–98
6. G. K. Kamenev, A. V. Lotov, “Approximation of the effective hull of a nonconvex multidimensional set given by a nonlinear mapping”, Dokl. Math., 97:1 (2018), 104–108
7. A. V. Lotov, “New external estimate for the reachable set of a nonlinear multistep dynamic system”, Comput. Math. Math. Phys., 58:2 (2018), 196–206
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