Outperforming the Best 1D Low-Discrepancy Constructions with a Greedy Algorithm

TL;DR

This paper demonstrates that a greedy algorithm can produce uniformly spread sequences on [0,1) that outperform traditional low-discrepancy sequences, with promising results in higher dimensions.

Contribution

It introduces a novel greedy approach for constructing low-discrepancy sequences that surpasses existing number-theoretic methods, supported by numerical experiments.

Findings

Greedy sequences outperform classical low-discrepancy sequences in one dimension.

The approach remains effective and regular in higher dimensions (2D and 3D).

Numerical results suggest potential for improved sequence design.

Abstract

The design of uniformly spread sequences on $[0,1)$ has been extensively studied since the work of Weyl and van der Corput in the early $20^{\text{th}}$ century. The current best sequences are based on the Kronecker sequence with golden ratio and a permutation of the van der Corput sequence by Ostromoukhov. Despite extensive efforts, it is still unclear if it is possible to improve these constructions further. We show, using numerical experiments, that a radically different approach introduced by Kritzinger in seems to perform better than the existing methods. In particular, this construction is based on a \emph{greedy} approach, and yet outperforms very delicate number-theoretic constructions. Furthermore, we are also able to provide the first numerical results in dimensions 2 and 3, and show that the sequence remains highly regular in this new setting.