Low-discrepancy point sets for non-uniform measures

TL;DR

This paper establishes the existence of low-discrepancy point sets for arbitrary non-uniform measures on the unit cube, improving previous bounds and enabling efficient numerical integration of discontinuous functions.

Contribution

The paper extends discrepancy theory to non-uniform measures, providing explicit bounds and connecting to cubature rules for faster integration of discontinuous functions.

Findings

Existence of low-discrepancy point sets with improved bounds

Upper bounds for discrepancy involving logarithmic factors

Application to numerical integration of discontinuous functions

Abstract

In the present paper we prove several results concerning the existence of low-discrepancy point sets with respect to an arbitrary non-uniform measure $\mu$ on the $d$-dimensional unit cube. We improve a theorem of Beck, by showing that for any $d \geq 1$, $N \geq 1,$ and any non-negative, normalized Borel measure $\mu$ on $[0,1]^d$ there exists a point set $x_1, \dots, x_N \in [0,1]^d$ whose star-discrepancy with respect to $\mu$ is of order $$ D_N^*(x_1, \dots, x_N; \mu) \ll \frac{(\log N)^{(3d+1)/2}}{N}. $$ For the proof we use a theorem of Banaszczyk concerning the balancing of vectors, which implies an upper bound for the linear discrepancy of hypergraphs. Furthermore, the theory of large deviation bounds for empirical processes indexed by sets is discussed, and we prove a numerically explicit upper bound for the inverse of the discrepancy for Vapnik--\v{C}ervonenkis classes. Finally, using a recent version of the Koksma--Hlawka inequality due to Brandolini, Colzani, Gigante and Travaglini, we show that our results imply the existence of cubature rules yielding fast convergence rates for the numerical integration of functions having discontinuities of a certain form.