Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/6097
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Title: Well conditioned spherical designs for integration and interpolation on the two-sphere
Authors: An, C
Chen, X 
Sloan, IH
Womersley, RS
Issue Date: 2010
Source: SIAM journal on numerical analysis, 2010, v. 48, no. 6, p. 2135–2157
Abstract: A set X[sub N] 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 X [sub N] 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 P [sub t] 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.
Keywords: Spherical design
Fundamental system
Mesh norm
Maximum determinant
Lebesgue constant
Numerical integration
Interpolation
Interval method
Publisher: Society for Industrial and Applied Mathematics
Journal: SIAM journal on numerical analysis 
ISSN: 0036-1429 (print)
1095-7170 (online)
DOI: 10.1137/100795140
Rights: © 2010 Society for Industrial and Applied Mathematics
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