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 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.
A new robust algorithm to estimate star discrepancy values has been proposed in @cite_10 . This algorithm has been reported to give very accurate discrepancy estimates, and our experiments confirm these statements. We thus use this algorithm for the intermediate discrepancy evaluations; i.e., for the optimization process of creating good candidate point configurations. Where feasible, we do a final evaluation of the candidate sets using the exact DEM-algorithm described above.
@cite_10
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We present a new algorithm for estimating the star discrepancy of arbitrary point sets. Similar to the algorithm for discrepancy approximation of Winker and Fang [SIAM J. Numer. Anal., 34 (1997), pp. 2028-2042] it is based on the optimization algorithm threshold accepting. Our improvements include, amongst others, a nonuniform sampling strategy, which is more suited for higher-dimensional inputs and additionally takes into account the topological characteristics of given point sets, and rounding steps which transform axis-parallel boxes, on which the discrepancy is to be tested, into critical test boxes. These critical test boxes provably yield higher discrepancy values and contain the box that exhibits the maximum value of the local discrepancy. We provide comprehensive experiments to test the new algorithm. Our randomized algorithm computes the exact discrepancy frequently in all cases where this can be checked (i.e., where the exact discrepancy of the point set can be computed in feasible time). Most importantly, in higher dimensions the new method behaves clearly better than all previously known methods.