A note on the CBC-DBD construction of lattice rules with general positive weights

TL;DR

This paper extends the CBC-DBD construction method for generating vectors of rank-1 lattice rules from product weights to general positive weights, enabling explicit construction for higher dimensions in quasi-Monte Carlo integration.

Contribution

It generalizes the CBC-DBD construction to arbitrary positive weights, addressing an open problem and broadening applicability for lattice rule design.

Findings

The generalized CBC-DBD method is effective for arbitrary positive weights.

The algorithm is competitive with classical CBC in the case of POD weights.

Explicit constructions are now possible for higher dimensions with general weights.

Abstract

Lattice rules are among the most prominently studied quasi-Monte Carlo methods to approximate multivariate integrals. A rank-$1$ lattice rule to approximate an $s$-dimensional integral is fully specified by its \emph{generating vector} $\boldsymbol{z} \in \mathbb{Z}^s$ and its number of points~$N$. While there are many results on the existence of ``good'' rank-$1$ lattice rules, there are no explicit constructions of good generating vectors for dimensions $s \ge 3$. This is why one usually resorts to computer search algorithms. In a recent paper by Ebert et al. in the Journal of Complexity, we showed a component-by-component digit-by-digit (CBC-DBD) construction for good generating vectors of rank-1 lattice rules for integration of functions in weighted Korobov classes. However, the result in that paper was limited to product weights. In the present paper, we shall generalize this result to arbitrary positive weights, thereby answering an open question posed in the paper of Ebert et al. We also include a short section on how the algorithm can be implemented in the case of POD weights, by which we see that the CBC-DBD construction is competitive with the classical CBC construction.