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

Abstract

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 R d . 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.