Component-by-component construction of good lattice rules

Abstract

This paper provides a novel approach to the construction of good lattice rules for the integration of Korobov classes of periodic functions over the unit $s$-dimensional cube. Theorems are proved which justify the construction of good lattice rules one component at a time—that is, the lattice rule for dimension $s+1$ is obtained from the rule for dimension $s$ by searching over all possible choices of the $(s+1)$th component, while keeping all the existing components unchanged. The construction, which goes against accepted wisdom, is illustrated by numerical examples. The construction is particularly useful if the components of the integrand are ordered, in the sense that the first component is more important than the second, and so on.