Lattice enumeration via linear programming

Abstract

Given a positive integer d and \({{\varvec{a}}}_{1},\dots ,{{\varvec{a}}}_{r}\) a family of vectors in \({{\mathbb {R}}}^d\), \(\{k_1{{\varvec{a}}}_{1}+\dots +k_r{{\varvec{a}}}_{r}: k_1,\dots ,k_r \in {{\mathbb {Z}}}\}\subset {{\mathbb {R}}}^d\) is the so-called lattice generated by the family. In high dimensional integration, prescribed lattices are used for constructing reliable quadrature schemes. The quadrature points are the lattice points lying on the integration domain, typically the unit hypercube \([0,1)^d\) or a rescaled shifted hypercube. It is crucial to have a cost-effective method for enumerating lattice points within such domains. Undeniably, the lack of such fast enumeration procedures hinders the applicability of lattice rules. Existing enumeration procedures exploit intrinsic properties of the lattice at hand, such as periodicity, orthogonality, recurrences, etc. In this paper, we unveil a general-purpose fast lattice enumeration algorithm based on linear programming (named FLE-LP).