Points on manifolds with asymptotically optimal covering radius

TL;DR

This paper extends the understanding of optimal point distributions on manifolds, showing that quasi-Monte Carlo points achieve asymptotically optimal covering radii on general compact smooth Riemannian manifolds, including the Grassmannian.

Contribution

It generalizes results from spheres to arbitrary compact smooth Riemannian manifolds and demonstrates the asymptotic optimality of quasi-Monte Carlo points in this broader setting.

Findings

Quasi-Monte Carlo points achieve asymptotically optimal covering radii on manifolds.

Theoretical bounds on covering radius extend from spheres to general manifolds.

Numerical experiments confirm the theoretical results on the Grassmannian manifold.

Abstract

Given a finite set of points on the Euclidean sphere, the worst case quadrature error in Sobolev spaces has recently been shown to provide upper bounds on the covering radius of the point set. Moreover, quasi-Monte Carlo integration points on the sphere achieve the asymptotically optimal covering radius. Here, we extend these results to points on compact smooth Riemannian manifolds and provide numerical experiments illustrating our findings for the Grassmannian manifold.