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

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 ${\mathbb{H}}^{s^{\prime }}(S^d)$ for all $s^{\prime }\in (d/2,s)$. This leads to the question of determining the supremum of such $s$, for which we provide estimates based on computations for the aforementioned sequences.

34 pages, 3 figures, 1 table

References in corpus (4)

• Point sets on the sphere ${\mathbb{S}}^2$ with small spherical cap discrepancy
• A simple Proof of Stolarsky's Invariance Principle
• Quasi-Monte Carlo rules for numerical integration over the unit sphere ${\mathbb{S}}^2$
• Quadrature rules and distribution of points on manifolds

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