QMC strength for some random configurations on the sphere

TL;DR

This paper investigates the average quasi-Monte Carlo (QMC) strength of certain random point configurations on the sphere, extending previous conjectures about well-known deterministic point sets.

Contribution

It provides an analysis of the average QMC strength for random configurations on the sphere, offering insights beyond deterministic point sets.

Findings

Average QMC strength for random configurations analyzed

Results support conjectures about deterministic point sets

Provides bounds and estimates for QMC strength on the sphere

Abstract

A sequence $(X_N ) \subset \mathbb S^d$ of N-point sets from the d-dimensional sphere has QMC strength $s^*>d/2$ if it has worst-case error of optimal order, $N^{s/d}$, for Sobolev spaces of order $s$ for all $d/2 < s < s^*$ , and the order is not optimal for $s > s^*$ . In arXiv:1208.3267 conjectured values of the strength are given for some well known point families in $\mathbb S^2$ based on numerical results. We study the average QMC strength for some related random configurations.