Systematic optimization methods for uniform designs under wrap-around L2-discrepancy

As a successful practice of the quasi-Monte Carlo method in computer experiments, uniform design aims to distribute points evenly on a restricted domain. Such a point set has low discrepancy and has enjoyed increasing popularity in applications. However, most designs obtained by the numerical optimization algorithms in literature are just nearly uniform, thus there is significant room for improvement. This paper reviews the existing work on uniform designs and then characterizes their structure under the wrap-around L2-discrepancy (WD). Deterministic construction methods for uniform designs under WD with any number of levels are theoretically proposed, which break through a common limitation of setting the level number to be a prime or a prime power. Based on the above, we provide a systematic optimization algorithm to search for uniform designs under WD with more general parameters for practical use. Numerical experiments show that the performance and runtime of the proposed algorithm are far superior to existing ones.