On the Behavior of the Weighted Star Discrepancy Bounds for Shifted Lattice Rules

Abstract

We examine the question of constructing shifted lattice rules of rank one with an arbitrary number of points n, an arbitrary shift, and small weighted star discrepancy. An upper bound on the weighted star discrepancy, that depends on the lattice parameters and is easily computable, serves as a figure of merit. It is known that there are lattice rules for which this upper bound converges as O(n −1+δ ) for any δ>0, uniformly over the shift, and lattice rules that achieve this convergence rate can be found by a component-by-component (CBC) construction. In this paper, we examine practical aspects of these bounds and results, such as: What is the shape of the probability distribution of the figure of merit for a random lattice with a given n? Is the CBC construction doing much better than just picking the best out of a few random lattices, or much better than using a randomized CBC construction that tries only a small number of random values at each step? How does the figure of merit really behave as a function of n for the best lattice, and on average for a random lattice, say for n under a million? Do we observe a convergence rate near O(n −1) in that range of values of n? Finally, is the figure of merit a tight bound on the true discrepancy, or is there a large gap between the two?