Numerical Integration over the Unit Sphere by using spherical t-design

TL;DR

This paper evaluates spherical t-designs for numerical integration over the unit sphere, demonstrating that well-conditioned designs outperform efficient designs in accuracy, especially for integrals with singularities, through extensive numerical experiments.

Contribution

It introduces and compares well-conditioned and efficient spherical t-designs, showing the superiority of WSTD in worst-case error and practical integral approximation, including singular integrands.

Findings

WSTD outperforms ESTD in worst-case error.

WSTD provides more accurate integral approximations, especially for singular functions.

Numerical results confirm the effectiveness of WSTD with various quadrature rules.

Abstract

This paper studies numerical integration over the unit sphere $ \mathbb{S}^2 \subset \mathbb{R}^{3} $ by using spherical $t$-design, which is an equal positive weights quadrature rule with polynomial precision $t$. We investigate two kinds of spherical $t$-designs with $t$ up to 160. One is well conditioned spherical $t$-design(WSTD), which was proposed by [1] with $ N=(t+1)^{2} $. The other is efficient spherical $t$-design(ESTD), given by Womersley [2], which is made of roughly of half cardinality of WSTD. Consequently, a series of persuasive numerical evidences indicates that WSTD is better than ESTD in the sense of worst-case error in Sobolev space $ \mathbb{H}^{s}(\mathbb{S}^2) $. Furthermore, WSTD is employed to approximate integrals of various of functions, especially including integrand has a point singularity over the unit sphere and a given ellipsoid. In particular, to deal with singularity of integrand, Atkinson's transformation [3] and Sidi's transformation [4] are implemented with the choices of `grading parameters' to obtain new integrand which is much smoother. Finally, the paper presents numerical results on uniform errors for approximating representive integrals over sphere with three quadrature rules: Bivariate trapezoidal rule, Equal area points and WSTD.