Component-by-Component Construction of Good Lattice Rules with a Composite Number of Points

Abstract

We develop algorithms to construct rank-1 lattice rules in weighted Korobov spaces of periodic functions and shifted rank-1 lattice rules in weighted Sobolev spaces of non-periodic functions. Analyses are given which show that the rules so constructed achieve strong QMC tractability error bounds. Unlike earlier analyses, there is no assumption that n, the number of quadrature points, be a prime number. However, we do assume that there is an upper bound on the number of distinct prime factors of n. The generating vectors and shifts characterizing the rules are constructed ‘component-by-component,’ that is, the ( d+1)th components of the generating vectors and shifts are obtained using one-dimensional searches, with the previous d components kept unchanged.