Well conditioned spherical designs for integration and interpolation on the two-sphere

Congpei An, Xiaojun Chen, Ian H. Sloan, Robert S. Womersley

Research output: Journal article publicationJournal articleAcademic researchpeer-review

20 Citations (Scopus)

Abstract

A set XN of N points on the unit sphere is a spherical t-design if the average value of any polynomial of degree at most t over XN is equal to the average value of the polynomial over the sphere. This paper considers the characterization and computation of spherical t-designs on the unit sphere S2 ⊂ ℝ3 when N ≥ (t + 1)2, the dimension of the space Pt of spherical polynomials of degree at most t. We show how to construct well conditioned spherical designs with N ≥ (t + 1) 2 points by maximizing the determinant of a matrix while satisfying a system of nonlinear constraints. Interval methods are then used to prove the existence of a true spherical t-design very close to the calculated points and to provide a guaranteed interval containing the determinant. The resulting spherical designs have good geometrical properties (separation and mesh norm). We discuss the usefulness of the points for both equal weight numerical integration and polynomial interpolation on the sphere and give an example.
Original languageEnglish
Pages (from-to)2135-2157
Number of pages23
JournalSIAM Journal on Numerical Analysis
Volume48
Issue number6
DOIs
Publication statusPublished - 1 Dec 2010

Keywords

  • Fundamental system
  • Interpolation
  • Interval method
  • Lebesgue constant
  • Maximum determinant
  • Mesh norm
  • Numerical integration
  • Spherical design

ASJC Scopus subject areas

  • Numerical Analysis

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