Comparison of probabilistic and deterministic point sets on the sphere

Peter Grabner, Tetiana Stepaniuk

Research output: Contribution to journalArticleResearchpeer-review

Abstract

In this paper we make a comparison between certain probabilistic and deterministic point sets and show that some deterministic constructions (especially spherical t-designs) are better or as good as probabilistic ones like the jittered sampling model. We find asymptotic equalities for the discrete Riesz s-energy of sequences of well separated t-designs on the unit sphere S d⊂R d+1, d≥2. The case d=2 was studied in Hesse (2009) and Hesse and Leopardi (2008). In Bondarenko et al., (2015) it was established that for d≥2, there exists a constant c d, such that for every N>c dt d there exists a well-separated spherical t-design on S d with N points. This paper gives results, based on recent developments that there exists a sequence of well separated spherical t-designs such that t and N are related by N≍t d.

LanguageEnglish
Pages128-143
Number of pages16
JournalJournal of approximation theory
Volume239
DOIs
StatusPublished - 2019

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T-designs
Point Sets
Unit Sphere
Equality
Sampling
Energy

Keywords

  • Discrete energy
  • Energy integral
  • Equal-area partition
  • Equal-weight numerical integration
  • Sphere
  • t-design
  • The s-energy
  • Well-separated point sets

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Mathematics(all)
  • Numerical Analysis

Cite this

Comparison of probabilistic and deterministic point sets on the sphere. / Grabner, Peter; Stepaniuk, Tetiana.

In: Journal of approximation theory, Vol. 239, 2019, p. 128-143.

Research output: Contribution to journalArticleResearchpeer-review

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