Shifted lattice rules based on a general weighted discrepancy for integrals over Euclidean space

摘要

We approximate weighted integrals over Euclidean space by using shifted rank-1 lattice rules with good bounds on the “generalised weighted star discrepancy”. This version of the discrepancy corresponds to the classic L∞ weighted star discrepancy via a mapping to the unit cube. The weights here are general weights rather than the product weights considered in earlier works on integrals over Rd. Known methods based on an averaging argument are used to show the existence of these lattice rules, while the component-by-component technique is used to construct the generating vector of these shifted lattice rules. We prove that the bound on the weighted star discrepancy considered here is of order O(n−1+δ) for any δ>0 and with the constant involved independent of the dimension. This convergence rate is better than the O(n−1/2) achieved so far for both Monte Carlo and quasi-Monte Carlo methods.