Constructing Low Star Discrepancy Point Sets with Genetic Algorithms

Geometric discrepancies are standard measures to quantify the irregularity of distributions. They are an important notion in numerical integration. One of the most important discrepancy notions is the so-called \emph{star discrepancy}. Roughly speaking, a point set of low star discrepancy value allows for a small approximation error in quasi-Monte Carlo integration. It is thus the most studied discrepancy notion. In this work we present a new algorithm to compute point sets of low star discrepancy. The two components of the algorithm (for the optimization and the evaluation, respectively) are based on evolutionary principles. Our algorithm clearly outperforms existing approaches. To the best of our knowledge, it is also the first algorithm which can be adapted easily to optimize inverse star discrepancies.Comment: Extended abstract appeared at GECCO 2013. v2: corrected 3 numbers in table

Heuristic Approaches to Obtain Low-Discrepancy Point Sets via Subset Selection

Building upon the exact methods presented in our earlier work [J. Complexity, 2022], we introduce a heuristic approach for the star discrepancy subset selection problem. The heuristic gradually improves the current-best subset by replacing one of its elements at a time. While we prove that the heuristic does not necessarily return an optimal solution, we obtain very promising results for all tested dimensions. For example, for moderate point set sizes $30 \leq n \leq 240$ in dimension 6, we obtain point sets with $L_{\infty}$ star discrepancy up to 35% better than that of the first $n$ points of the Sobol' sequence. Our heuristic works in all dimensions, the main limitation being the precision of the discrepancy calculation algorithms. We also provide a comparison with a recent energy functional introduced by Steinerberger [J. Complexity, 2019], showing that our heuristic performs better on all tested instances

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum