QMC designs: optimal order Quasi Monte Carlo Integration schemes on the sphere

TL;DR

This paper introduces the concept of QMC designs on the sphere, providing methods for their generation, analyzing their properties, and demonstrating their effectiveness for numerical integration in Sobolev spaces.

Contribution

It defines QMC designs for the sphere, links them to spherical designs and energy minimizers, and shows many known point sets are effective QMC schemes for certain smoothness levels.

Findings

Minimizers of energy for the reproducing kernel form QMC designs.

Many classical point sets on the sphere are QMC designs for specific smoothness.

Random points do not form QMC designs for any smoothness level.

Abstract

We study equal weight numerical integration, or Quasi Monte Carlo (QMC) rules, for functions in a Sobolev space $H^s(S^d)$ with smoothness parameter $s>d/2$ defined over the unit sphere $S^d$ in $R^{d+1}$. Focusing on $N$-point sets that achieve optimal order QMC error bounds (as is the case for efficient spherical designs), we are led to introduce the concept of QMC designs: these are sequences of $N$-point node sets $X_N$ on $S^d$ such that the worst-case error of the corresponding QMC rules satisfy a bound of order $O(N^{-s/d})$ as $N\to\infty$ with an implied constant that depends on the $H^s(S^d)$-norm. We provide methods for generation and numerical testing of QMC designs. As a consequence of a recent result of Bondarenko et al. on the existence of spherical designs with appropriate number of points, we show that minimizers of the $N$-point energy for the reproducing kernel for $H^s(S^d)$, $s>d/2$, form a sequence of QMC designs for $H^s(S^d)$. Furthermore, without appealing to the Bondarenko et al. result, we prove that point sets that maximize the sum of suitable powers of the Euclidean distance between pairs of points form a sequence of QMC designs for $H^s(S^d)$ with $s\in(d/2,d/2+1)$. Numerical experiments suggest that many familiar sequences of point sets on the sphere (equal area, spiral, minimal [Coulomb or log.] energy, and Fekete points) are QMC designs for appropriate values of $s$. For comparison purposes we show that sets of random points that are independently and uniformly distributed on the sphere do not constitute QMC designs for any $s>d/2$. If $(X_N)$ is a sequence of QMC designs for $H^s(S^d)$, we prove that it is also a sequence of QMC designs for $\mathbb{H}^{s'}(S^d)$ for all $s'\in(d/2,s)$. This leads to the question of determining the supremum of such $s$, for which we provide estimates based on computations for the aforementioned sequences.