High-Dimensional Quasi-Monte Carlo via Combinatorial Discrepancy

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.