On theL2-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 lowdiscrepancy sets. Finally, we present a fairly precise formula for the performance of a recent algorithm due to Heinrich for computing the L 2 -discrepancy. 1998Academic Press

Parallel Quasi-Monte Carlo Integration by Partitioning Low Discrepancy Sequences

“…2.4]. While this coincides with our results, the derivation in Section 3.2 is more general, as it includes scrambled radical inverses [15] (see Section 3.3), which do not result in equidistant step size for leapfrogging.…”

Random and Deterministic Digit Permutations of the Halton Sequence

“…Moreover, as was observed by Matoušek [10], if N is small and s is large, the L 2 -discrepancy of any point set is close to the best possible L 2 -discrepancy.…”

Randomized Quasi-Monte Carlo: An Introduction for Practitioners

“…Doing permutations like this for 31 bits or more would be very time-consuming, but in fact one can do it for the first k bits only, and then generate the other bits randomly and independently across points and coordinates. From a statistical viewpoint this is equivalent to NUS and less time-consuming [48]. NUS also works in general base b: for each digit of each coordinate, we generate a random permutation of b elements to permute the points according to this digit.…”

“…One popular example is the (left) linear matrix scramble [48,64,56]: left-multiply each matrix C j by a random w × w matrix M j , non-singular and lower triangular, mod b. With this scrambling just by itself, each point does not have the uniform distribution (e.g., the point 0 is unchanged), but one can apply a random digital shift in base b after the matrix scramble to obtain an RQMC scheme.…”

“…Stratification and other RQMC methods based for example on lattice rules and digital nets can improve the rate; see Section 2. Some RQMC methods provide a better convergence rate than the squared worst-case error, because the average over the randomizations is sometimes better than the worst case [17,48,56]. Note that because of the nonzero covariances, we cannot use the sample variance of the f (U i ) to estimate Var[μ n,rqmc ] as for MC.…”