On Korobov Lattice Rules in Weighted Spaces

Abstract

This paper studies the error bounds of multivariate integration in weighted function spaces using lattice rules of the Korobov form, in which the generating vector for an n-point rule with n prime has the form (1,a,...,ad -1) (mod,n). With the parameter a chosen optimally, we establish new error bounds for Korobov lattice rules in weighted Korobov spaces. In particular, we prove that if the weights decay sufficiently fast, the optimal Korobov lattice rule has an error bound of order O(n^{-\alpha/2 +\delta})$ (for arbitrary $\delta \,{>}\, 0$), with the implied constant depending at worst polynomially on the dimension. Here $\alpha \,{>}\,1$ is the smoothness parameter of the weighted Korobov spaces. We generalize the construction to the case where n is a product of arbitrary distinct prime numbers, with the purpose of reducing the construction cost without sacrificing much of the quality of the lattice rules. A corresponding result is deduced for weighted Sobolev spaces of nonperiodic functions, using randomly shifted optimal Korobov lattice rules. A comparison of the worst-case errors for Korobov lattice rules and the recent component-by-component constructions is presented. The investigation establishes the usefulness of (shifted) optimal Korobov lattice rules for integration, even in high dimensions, if the weights which characterize the weighted spaces are suitably chosen.