Quasi-Monte Carlo rules for numerical integration over the unit sphere $\mathbb{S}^2$

TL;DR

This paper develops and analyzes a quasi-Monte Carlo method for numerical integration on the unit sphere using lifted $(0,m,2)$-nets, demonstrating near-optimal discrepancy properties and improved bounds over previous methods.

Contribution

It introduces a novel lifting approach for $(0,m,2)$-nets to the sphere and proves their near-optimal discrepancy and uniform distribution properties.

Findings

Construction is nearly optimal for spherical rectangle discrepancy.

Point sets are asymptotically uniformly distributed on $S^2$.

Upper bound on spherical cap $L_2$-discrepancy of order $N^{-1/2} (\log N)^{1/2}$.

Abstract

We study numerical integration on the unit sphere $\mathbb{S}^2 \subset \mathbb{R}^3$ using equal weight quadrature rules, where the weights are such that constant functions are integrated exactly. The quadrature points are constructed by lifting a $(0,m,2)$-net given in the unit square $[0,1]^2$ to the sphere $\mathbb{S}^2$ by means of an area preserving map. A similar approach has previously been suggested by Cui and Freeden [SIAM J. Sci. Comput. 18 (1997), no. 2]. We prove three results. The first one is that the construction is (almost) optimal with respect to discrepancies based on spherical rectangles. Further we prove that the point set is asymptotically uniformly distributed on $\mathbb{S}^2$. And finally, we prove an upper bound on the spherical cap $L_2$-discrepancy of order $N^{-1/2} (\log N)^{1/2}$ (where $N$ denotes the number of points). This slightly improves upon the bound on the spherical cap $L_2$-discrepancy of the construction by Lubotzky, Phillips and Sarnak [Comm. Pure Appl. Math. 39 (1986), 149--186]. Numerical results suggest that the $(0,m,2)$-nets lifted to the sphere $\mathbb{S}^2$ have spherical cap $L_2$-discrepancy converging with the optimal order of $N^{-3/4}$.