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Iterative method for constructing coverings of the multidimensional unit sphere

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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.

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Authors and Affiliations

  1. Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333, Russia

    G. K. Kamenev

  2. Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119991, Russia

    A. V. Lotov & T. S. Mayskaya

Authors
  1. G. K. Kamenev
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  2. A. V. Lotov
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  3. T. S. Mayskaya
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Correspondence to G. K. Kamenev.

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Original Russian Text © G.K. Kamenev, A.V. Lotov, T.S. Mayskaya, 2013, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2013, Vol. 53, No. 2, pp. 181–194.

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Kamenev, G.K., Lotov, A.V. & Mayskaya, T.S. Iterative method for constructing coverings of the multidimensional unit sphere. Comput. Math. and Math. Phys. 53, 131–143 (2013). https://doi.org/10.1134/S0965542513020085

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