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On the L2-discrepancy for anchored boxes

The L2-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 points in dimension 30) then nearly the lowest possible L2 -discrepancy is attained by a pathological point set, and hence the L2 -discrepancy may not be very relevant for relatively small sets. Recently, Hickernell obtained a formula for the expected L2-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 lowdiscrepancy sets. Finally, we present a fairly precise formula for the performance of a recent algorithm due to Heinrich for computing the L2-discrepancy. 1998