From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules

TL;DR

This paper surveys the development of van der Corput sequences and their generalizations, highlighting their importance in quasi-Monte Carlo methods for numerical integration from their origins to modern constructions.

Contribution

It provides a comprehensive overview of the evolution of van der Corput sequences and introduces recent advances in sequence constructions for quasi-Monte Carlo algorithms.

Findings

Historical development of van der Corput sequences

Connections to modern quasi-Monte Carlo sequences

Insights into sequence properties and applications

Abstract

In 1935 J.G. van der Corput introduced a sequence which has excellent uniform distribution properties modulo 1. This sequence is based on a very simple digital construction scheme with respect to the binary digit expansion. Nowadays the van der Corput sequence, as it was named later, is the prototype of many uniformly distributed sequences, also in the multi-dimensional case. Such sequences are required as sample nodes in quasi-Monte Carlo algorithms, which are deterministic variants of Monte Carlo rules for numerical integration. Since its introduction many people have studied the van der Corput sequence and generalizations thereof. This led to a huge number of results. On the occasion of the 125th birthday of J.G. van der Corput we survey many interesting results on van der Corput sequences and their generalizations. In this way we move from van der Corput's ideas to the most modern constructions of sequences for quasi-Monte Carlo rules, such as, e.g., generalized Halton sequences or Niederreiter's $(t,s)$-sequences.