On the L2-Discrepancy for Anchored Boxes

Abstract The L 2 -discrepancy for anchored axis-parallel boxes has been used in several recent computational studies, mostly related to numerical integration, as a measure of the quality of uniform distribution of a given point set. We point out that if the number of points is not large enough in terms of the dimension (e.g., fewer than 10 4 points in dimension 30) then nearly the lowest possible L 2 -discrepancy is attained by a pathological point set, and hence the L 2 -discrepancy may not be very relevant for relatively small sets. Recently, Hickernell obtained a formula for the expected L 2 -discrepancy of certain randomized low-discrepancy set constructions introduced by Owen. We note that his formula remains valid also for several modifications of these constructions which admit a very simple and efficient implementation. We also report results of computational experiments with various constructions of low-discrepancy sets. Finally, we present a fairly precise formula for the performance of a recent algorithm due to Heinrich for computing the L 2 -discrepancy.

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