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

Abstract

Spherical t-designs are point sets $$X_{M}=\{x_{1},\ldots, x_{M}\} \subset {\mathbb{S}^{2}}$$ which provide quadrature rules with equal weights for the sphere which are exact for polynomials up to degree t. In this paper we consider the problem of finding numerical spherical t-designs on the sphere $${\mathbb{S}^{2}}$$ for high polynomial degree $${t \in {\mathbb{N}}}$$. That is, we compute numerically local minimizers of a certain quadrature error A t (X M ). The quadrature error A t was also used for a variational characterization of spherical t-designs by Sloan and Womersley (J Approx Theory 159:308–318, 2009). For the minimization problem 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 $${\mathcal {O}(t^{2}\log t + M \log^{2}(1/\varepsilon))}$$ arithmetic operations, where $${\varepsilon >0 }$$ is a prescribed accuracy. Using these methods we present numerical spherical t-designs for t ≤ 1,000, even in the case $${M\approx \frac{1}{2} t^{2}}$$.