An improved Halton sequence for implementation in quasi-Monte Carlo methods

TL;DR

This paper introduces an improved Halton sequence designed to enhance distribution properties in high dimensions for quasi-Monte Carlo methods, along with a novel scrambling algorithm for irrational digital sequences, demonstrating empirical performance gains.

Contribution

It presents a new adapted Halton sequence using irrational-based van der Corput sequences and introduces the first scrambling algorithm for irrational digital sequences.

Findings

Empirical improvements in numerical integration accuracy.

Enhanced distribution properties in high-dimensional settings.

First implementation of a scrambling algorithm for irrational sequences.

Abstract

Despite possessing the low-discrepancy property, the classical d dimensional Halton sequence is known to exhibit poorly distributed projections when d becomes even moderately large. This, in turn, often implies bad performance when implemented in quasi-Monte Carlo (QMC) methods in comparison to, for example, the Sobol' sequence. As an attempt to eradicate this issue, we propose an adapted Halton sequence built by integer and irrational based van der Corput sequences and show empirically improved performance with respect to the accuracy of estimates in numerical integration and simulation. In addition, for the first time, a scrambling algorithm is proposed for irrational based digital sequences.