In Search for Good Chebyshev Lattices

Abstract

Recently we introduced a new framework to describe some point sets used for multivariate integration and approximation (Cools and Poppe, BIT Numer Math 51:275–288, 2011), which we called Chebyshev lattices. The associated integration rules are equal weight rules, with corrections for the points on the boundary. In this text we detail the development of exhaustive search algorithms for good Chebyshev lattices where the cost of the rules, i.e., the number of points needed for a certain degree of exactness, is used as criterium. Almost loopless algorithms are considered to avoid dependencies on the rank of the Chebyshev lattice and the dimension. Also, several optimisations are applied: reduce the vast search space by exploiting symmetries, lower the cost of the point set creation and minimise the cost of the degree verification. The concluding summary of the search results indicates that higher rank rules in general are better and that the blending formulae due to Godzina lead to the best rules within the class of Chebyshev lattice rules: no better rules have been found in the searches conducted in up to five dimensions.