A construction of higher-rank lattice rules

Lattice rules are quasi-Monte Carlo methods for numerical multiple integration that are based on the selection of an s-dimensional integration lattice. The abscissa set is the intersection of the integration lattice with the unit hypercube. It is well-known that the abscissa set of a lattice rule can be generated by a number of fixed rational vectors. In general, different sets of generators produce different integration lattices and rules, and a given rule has many different generator sets. The rank of the rule is the minimum number of generators required to span the abscissa set. A lattice rule is usually specified by a generator set, and the quality of the rule varies with the choice of generator set. This paper describes a new method for the construction of generator sets for higher-rank rules that is based on techniques arising from the theory of simultaneous Diophantine approximation. The method extends techniques currently applied in the rank 1 case. © 2001 IMACS. Published by Elsevier Science B.V. All rights reserved