Spherical T-designs for approximations on the sphere

Yang Zhou, Xiaojun Chen

Research output: Journal article publicationJournal articleAcademic researchpeer-review

6 Citations (Scopus)

Abstract

A spherical t-design is a set of points on the unit sphere that are nodes of a quadrature rule with positive equal weights that is exact for all spherical polynomials of degree ≤ t. The existence of a spherical t-design with (t+1)2 points in a set of interval enclosures on the unit sphere S2 ⊂ ℝ3 for any 0 ≤ t ≤ 100 is proved by Chen, Frommer, and Lang (2011). However, how to choose a set of points from the set of interval enclosures to obtain a spherical t-design with (t + 1)2 points is not given in loc. cit. It is known that (t + 1)2 is the dimension of the space of spherical polynomials of degree at most t in 3 variables on S2. In this paper we investigate a new concept of point sets on the sphere named spherical t-design (0 ≤ ∈ < 1), which are nodes of a positive, but not necessarily equal, weight quadrature rule exact for polynomials of degree ≤ t. The parameter ∈ is used to control the variation of the weights, while the sum of the weights is equal to the area of the sphere. A spherical t-design is a spherical t-design when ∈ = 0, and a spherical t-design is a spherical t- design for any 0 < ∈ < 1. We show that any point set chosen from the set of interval enclosures (loc. cit.) is a spherical t-design. We then study the worstcase error in a Sobolev space for quadrature rules using spherical t-designs, and investigate a model of polynomial approximation with l1-regularization using spherical t-designs. Numerical results illustrate the good performance of spherical t-designs for numerical integration and function approximation on the sphere.

Original languageEnglish
Pages (from-to)2831-2855
Number of pages25
JournalMathematics of Computation
Volume87
Issue number314
DOIs
Publication statusPublished - 5 Feb 2018

Keywords

  • Interval analysis
  • L-regularization
  • Numerical integration
  • Polynomial approximation
  • Spherical t-designs

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

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