Optimal Non-adaptive Approximation of Convex Bodies by Polytopes
In this paper we consider the problem of constructing numerical algorithms for approximating of convex compact bodies in d-dimensional Euclidean space by polytopes with any given accuracy. It is well known that optimal with respect to the order algorithms produce polytopes for which the accuracy in Hausdorff metric is inversely proportional to the number of vertices (faces) in the degree of 2∕(d − 1). Numerical approximation algorithms can be adaptive (active) when the vertices or faces are constructed successively, depending on the information obtained in the process of approximation, and non-adaptive (passive) when parameters of algorithms are defined on the basis of a priory information available. Approximation algorithms differ in the use of operations applied to the approximated body. Most common are indicator, support and distance (Minkowski) functions calculations. Some optimal active algorithms for arbitrary bodies approximation are known using support or distance function calculation operation. Optimal passive algorithms for smooth bodies approximation are known using support function calculation operation and extremal curvature information. It is known that there are no optimal non-adaptive algorithms for arbitrary bodies approximation using support function calculation operation. We consider optimal non-adaptive algorithms for arbitrary bodies approximation using projection function calculation operation.
This work was partially supported by the Russian Science Foundation, project no. 18-01-00465a.
- 1.Gruber, P.M.: Aspects of approximation of convex bodies. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry. Ch. 1.10, pp. 321–345. Elsevier Sci. Publishers B.V., Amsterdam (1993)Google Scholar
- 3.Lotov, A.V., Bushenkov, V.A., Kamenev, G.K.: Interactive decision maps. In: Approximation and Visualization of Pareto Frontier. Appl. Optimization, vol. 89. Kluwer Academic Publishers, Boston (2004)Google Scholar
- 4.Komp’yuter i poisk kompromissa. Metod dostizhimyh celej. Nauka, Moscow [The Feasible Goals Method. Nauka, Moscow]Google Scholar
- 6.Kamenev, G.K.: Optimal’nye adaptivnye metody poliedral’noj approksimacii vypuklyh tel. Izd. VC RAN, Moscow [Optimal Adaptive Methods for Polyhedral Approximation of Convex Bodies. CC of RAS, Moscow]Google Scholar
- 7.Vasil’yev, N.S.: On nonimprovable bounds of approximation of strongly convex bodies. Vopr. Kiber. 136, 49–56 (1988) [in Russian]Google Scholar
- 8.Dzholdybaeva, S.M., Kamenev, G.K.: Experimental Analysis of the Approximation of Convex Bodies by Polyhedra. Computing Centre of the USSR Academy of Sciences, Moscow (1991) [in Russian]Google Scholar
- 9.Sonnevend, G.: Asymptotically optimal, sequential methods for the approximation of convex, compact sets in Rn in the Hausdorff metrics. Colloq. Math. Soc. Janos Bolyai 35(2), 1075–1089 (1980)Google Scholar
- 13.Mayskaya, T.S.: Estimation of the radius of covering of the multidimensional unit sphere by metric net generated by spherical system of coordinates. In: Collected Papers of Young Scientists from the Faculty of Computational Mathematics and Cybernetics of Moscow State University. Vychisl. Mat. Kibern. Mosk. Gos. Univ., Moscow., No. 8, pp. 83–98 (2011). [in Russian]Google Scholar
- 16.Kamenev, G.K.: Polyhedral approximation of the ball by the deep holes method with an optimal growth order of the facet structure cardinality. In: Proceedings of International Conference on Numerical Geometry, Mesh Generation, and High-Performance Computations (NUMGRID2010), Moscow, October 11–13, 2010. Folium, Moscow, pp. 47–52 (2010)Google Scholar