Constructing lattice rules based on weighted degree of exactness and worst case error

Abstract

Recall that an integration rule is said to have a trigonometric degree of exactness m if it integrates exactly all trigonometric polynomials of degree ≤ m. In this paper we focus on high dimensions, say, d ≫ 6. We introduce three notions of weighted degree of exactness, where we use weights to characterize the anisotropicness of the integrand with respect to successive coordinate directions. Unlike in the classical unweighted setting, the minimal number of integration points needed to achieve a prescribed weighted degree of exactness no longer grows exponentially with d provided that the weights decay sufficiently fast. We present a component-by-component algorithm for the construction of a rank-1 lattice rule such that (i) it has a prescribed weighted degree of exactness, and (ii) its worst case error achieves the optimal rate of convergence in a weighted Korobov space. Then we introduce a modified, more practical, version of this algorithm which maximizes the weighted degree of exactness in each step of the construction. Both algorithms are illustrated by numerical results.