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 $$\mathbb {S}^d$$ . As by-products, we know the spherical ensemble, a well-studied DPP on $$\mathbb {S}^2$$ , generates asymptotically on average QMC design sequences for Sobolev space over $$\mathbb {S}^2$$ with smoothness $$1< s < 2$$ . Moreover, compared to i.i.d. uniform random points, we also know harmonic ensembles on $$\mathbb {S}^d$$ for $$d \ge 2$$ , which are DPPs defined by reproducing kernels for polynomial spaces over $$\mathbb {S}^d$$ , generate on average faster convergent sequences of the square worst-case error for Sobolev space over $$\mathbb {S}^d$$ with smoothness $$d/2+1/2< s < d/2+1$$ .