QMC designs and determinantal point processes

Abstract

In this paper, we deal with two types of determinantal point processes (DPPs) for equal weight numerical integration (quasi-Monte Carlo) rules on the sphere, and discuss the behavior of the worst-case numerical integration error for functions from Sobolev space over the d-dimensional unit sphere Sd. As by-products, we know the spherical ensemble, a well-studied DPP on S2, generates asymptotically on average QMC design sequences for Sobolev space over S2 with smoothness 1 < s< 2. Moreover, compared to i.i.d. uniform random points, we also know harmonic ensembles on Sd for d≥ 2, which are DPPs defined by reproducing kernels for polynomial spaces over Sd, generate on average faster convergent sequences of the square worst-case error for Sobolev space over Sd with smoothness d/ 2 + 1 / 2 <