High-Dimensional Quasi-Monte Carlo via Combinatorial Discrepancy

TL;DR

This paper develops high-dimensional Quasi-Monte Carlo methods using combinatorial discrepancy, providing theoretical error bounds and empirical performance assessments for functions with low effective dimension.

Contribution

It extends combinatorial discrepancy techniques to construct high-dimensional QMC point sets with proven error bounds in weighted spaces.

Findings

Error bounds established for high-dimensional QMC constructions.

Numerical experiments demonstrate practical effectiveness.

Performance is favorable for functions with low effective dimension.

Abstract

Monte Carlo (MC) and Quasi-Monte Carlo (QMC) methods are classical approaches for the numerical integration of functions $f$ over $[0,1]^d$. While QMC methods can achieve faster convergence rates than MC in moderate dimensions, their tractability in high dimensions typically relies on additional structure -- such as low effective dimension or carefully chosen coordinate weights -- since worst-case error bounds grow prohibitively large as $d$ increases. In this work, we study the construction of high-dimensional QMC point sets via combinatorial discrepancy, extending the recent QMC method of Bansal and Jiang. We establish error bounds for these constructions in weighted function spaces, and for functions with low effective dimension in both the superposition and truncation sense. We also present numerical experiments to empirically assess the performance of these constructions.