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

TL;DR

The paper tackles the degradation of projection quality in the classical Halton sequence for high-dimensional quasi-Monte Carlo methods. It introduces an interlaced Halton construction that inserts irrational-base van der Corput coordinates, specifically using an irrational base $\gamma$ tied to parameters $(p,q)$, between consecutive integer bases, and pairs this with a scrambling strategy in base $(p+1)$ to enable randomized QMC. The authors provide an algorithmic construction (Algorithm 1), a conjectured property of complete quasi-equidistribution for the scrambled irrational-base coordinates, and extensive empirical evidence showing improved accuracy over the classical Halton sequence—and competitive performance with Sobol'—in high-dimensional numerical integration and a computational-finance Asian option pricing task. This work offers a practical, scalable enhancement to QMC methods by improving projection uniformity and providing a feasible scrambling approach for irrational-based sequences, with potential broad impact on high-dimensional integration in science and finance.

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.

Figures (4)

• Figure 1: Projection of the first $500$ points of the $26^{th}$ and $27^{th}$ of the classical Halton sequence (bases 101 and 103) and the interlaced Halton sequence (bases $19.462..$ and $20.05..$)
• Figure 2: Integration error of function $f_1$. (Top) $a_j = j$ (Bottom) $a_j = j^2$.
• Figure 3: Integration error of function $f_2$. (Top) $c=1$ (Bottom) $c=0.1$.
• Figure 4: Absolute error (top) and estimated variance (bottom) for Asian option problem. (Left) $d=50$ (Right) $d=100$.

Theorems & Definitions (7)

• Definition 1
• Definition 2
• Definition 3: Definition 8.2, KIRKLEMIEUXWIART2023
• Conjecture 4
• Definition 5: Definition 4.1, DONGLEMIEUX2022
• Definition 6: Definition 4.4, DONGLEMIEUX2022
• Conjecture 7