In Search for Good Chebyshev Lattices
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.
This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State Science Policy Office. The scientific responsibility rests with its authors.
The authors acknowledge the support for this project by the Bijzonder Onderzoeksfonds of the Katholieke Universiteit Leuven.
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