Manuel Gräf, Daniel Potts: On the computation of spherical designs by a new optimization approach based on fast spherical Fourier transforms

Author(s):

Manuel Gräf
Daniel Potts

Title:

On the computation of spherical designs by a new optimization approach based on fast spherical Fourier transforms

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 12, 2010

Mathematics Subject Classification:

65T40	[ ]
65K10	[]
53B21	[]
49M15	[]
33C55	[]

Abstract:

In this paper we consider the problem of finding numerical spherical t-designs on the sphere S2. Spherical t-designs are point sets {x_1,...,x_M} of S2 which provide quadrature rules with equal weights for the sphere which are exact for polynomials up to degree t. We use a variational characterization of spherical t-designs proposed by Sloan and Womersley, where a minimization problem has to be solved. Therefor we regard several nonlinear optimization methods on manifolds, like Newton and conjugate gradient methods. We show that by means of the nonequispaced fast spherical Fourier transforms we perform gradient and Hessian evaluations in O(t2 log(t) + M log2(1\epsilon)) arithmetic operations. Using this we are able to compute spherical t-designs for t <= 1000 and present results even in the case M about (t2)/2.

Keywords:

spherical designs, variational characterization, optimization methods on Riemannian manifolds, spherical harmonics, iterative methods, nonequispaced Fourier methods on the sphere

Language:

English

Publication time:

07/2010