Uniformly distributed sequences generated by a greedy minimization of the $L_2$ discrepancy

TL;DR

This paper introduces greedy algorithms for generating uniformly distributed sequences in multi-dimensional spaces by minimizing $L_2$ discrepancy, with theoretical analysis and numerical validation showing their effectiveness and rational structure in one dimension.

Contribution

The paper presents novel greedy algorithms that produce uniformly distributed sequences by minimizing $L_2$ discrepancy, including explicit rational formulas in one dimension.

Findings

Sequences have excellent distribution properties.

In one dimension, sequences are rational and explicitly describable.

Discrepancy depends on the dimension $d$.

Abstract

The aim of this paper is to develop greedy algorithms which generate uniformly distributed sequences in the $d$-dimensional unit cube $[0,1]^d$. The figures of merit are three different variants of $L_2$ discrepancy. Theoretical results along with numerical experiments suggest that the resulting sequences have excellent distribution properties. The approach we follow here is motivated by recent work of Steinerberger and Pausinger who consider similar greedy algorithms, where they minimize functionals that can be related to the star discrepancy or energy of point sets. In contrast to many greedy algorithms where the resulting elements of the sequence can only be given numerically, we will find that in the one-dimensional case our algorithms yield rational numbers which we can describe precisely. In particular, we will observe that any initial segment of a sequence in $[0,1)$ can be naturally extended to a uniformly distributed sequence where all subsequent elements are of the form $x_N=\frac{2n-1}{2N}$ for some $n\in\{1,\dots,N\}$. We will also investigate the dependence of the $L_2$ discrepancy of the resulting sequences on the dimension $d$.