Geodesic
Iterative method for constructing coverings of the multidimensional unit sphere
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 2, pp. 181-194 
G. K. Kamenev; A. V. Lotov; T. S. Mayskaya. Iterative method for constructing coverings of the multidimensional unit sphere. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 2, pp. 181-194. http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_2_a1/
@article{ZVMMF_2013_53_2_a1,
     author = {G. K. Kamenev and A. V. Lotov and T. S. Mayskaya},
     title = {Iterative method for constructing coverings of the multidimensional unit sphere},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {181--194},
     year = {2013},
     volume = {53},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_2_a1/}
}
TY  - JOUR
AU  - G. K. Kamenev
AU  - A. V. Lotov
AU  - T. S. Mayskaya
TI  - Iterative method for constructing coverings of the multidimensional unit sphere
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2013
SP  - 181
EP  - 194
VL  - 53
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_2_a1/
LA  - ru
ID  - ZVMMF_2013_53_2_a1
ER  - 
%0 Journal Article
%A G. K. Kamenev
%A A. V. Lotov
%A T. S. Mayskaya
%T Iterative method for constructing coverings of the multidimensional unit sphere
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2013
%P 181-194
%V 53
%N 2
%U http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_2_a1/
%G ru
%F ZVMMF_2013_53_2_a1

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mischenko E. F., Matematicheskaya teoriya optimalnykh protsessov, Nauka, M., 1968

[2] Lotov A. V., Bushenkov V. A., Kamenev G. K., Chernykh O. L., Kompyuter i poisk kompromissa. Metod dostizhimykh tselei, Nauka, M., 1997

[3] Lotov A. V., Bushenkov V. A., Kamenev G. K., Interactive Decision Maps. Approximation and Visualization of Pareto Frontier, Kluwer, Boston, 2004 | MR | Zbl

[4] Hardin R. H., Sloane N. J. A., Smith W. D., “Spherical Codes”, A library of putatively optimal coverings of the sphere with $n$ equal caps, book in preparation http://neilsloane.com/coverings/index.html

[5] Conway J. H., Sloane N. J. A., Sphere packings, lattices and groups, Third Edition, Springer, Berlin, 1999 | MR

[6] Kamenev G. K., “Poliedralnaya approksimatsiya shara Metodom Glubokikh Yam s optimalnym poryadkom rosta moschnosti grannoi struktury”, Chislennaya geometriya, postroenie raschetnykh setok i vysokoproizvoditelnye vychisleniya, Tr. Mezhdunar. konf. NUMGRID2010 (Moskva, 11–13 oktyabrya 2010 g.), Izd. Folium, M., 2010, 47–52

[7] Gruber P. M., “Approximation of convex bodies”, Convexity and its applications, Birkhauser, Basel, 1983 | MR | Zbl

[8] Kamenev G. K., Optimalnye adaptivnye metody poliedralnoi approksimatsii vypuklykh tel, Izd. VTs RAN, M., 2007 | MR

[9] Kamenev G. K., Chislennoe issledovanie effektivnosti metodov poliedralnoi approksimatsii vypuklykh tel, Izd. VTs RAN, M., 2010

[10] Bushenkov V. A., Lotov A. V., Metody postroeniya i ispolzovaniya obobschennykh mnozhestv dostizhimosti, VTs AN SSSR, M., 1982

[11] Efremov R. V., Kamenev G. K., “Ob optimalnom poryadke rosta chisla vershin i gipergranei v klasse khausdorfovykh metodov poliedralnoi approksimatsii vypuklykh tel”, Zh. vychisl. matem. i matem. fiz., 51:6 (2011), 1018–1031 | MR | Zbl

[12] Rodzhers K., Ukladki i pokrytiya, Mir, M., 1968 | MR

[13] Schneider R., “Zur optimalen Approximation konvexer Hyperflächen durch Polyeder”, Math. Ann., 256:3 (1981), 289–301 | DOI | MR | Zbl

[14] http://www.ccas.ru/mmes/mmeda/mcdm.htm

[15] Maiskaya T. S., “Otsenka radiusa pokrytiya mnogomernoi edinichnoi sfery metricheskoi setyu, porozhdennoi sfericheskoi sistemoi koordinat”, Sb. statei molodykh uchenykh fakulteta VMK MGU, 8, Izd. VMK MGU, M., 2011, 83–98

[16] Lotov A. V., Maiskaya T. S., “Neadaptivnye metody poliedralnoi approksimatsii obolochki Edzhvorta–Pareto, ispolzuyuschie suboptimalnye metricheskie seti na sfere napravlenii”, Zh. vychisl. matem. i matem. fiz., 52:1 (2012), 35–47 | MR | Zbl

[17] Kamenev G. K., Lotov A. V., Maiskaya T. S., “Postroenie suboptimalnykh pokrytii mnogomernoi edinichnoi sfery”, Dokl. AN, 444:2 (2012), 153–155 | MR | Zbl