Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules

Authors:
Jan Baldeaux;Josef Dick;Gunther Leobacher;Dirk Nuyens;Friedrich Pillichshammer
Affiliations:
School of Finance and Economics, The University of Technology, Sydney, Australia 2007;School of Mathematics and Statistics, The University of New South Wales, Sydney, Australia 2052;Institut für Finanzmathematik, Universität Linz, Linz, Austria 4040;Department of Computer Science, K.U.Leuven, Heverlee, Belgium 3001;Institut für Finanzmathematik, Universität Linz, Linz, Austria 4040
Venue:
Numerical Algorithms
Year:
2012

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Abstract

We show how to obtain a fast component-by-component construction algorithm for higher order polynomial lattice rules. Such rules are useful for multivariate quadrature of high-dimensional smooth functions over the unit cube as they achieve the near optimal order of convergence. The main problem addressed in this paper is to find an efficient way of computing the worst-case error. A general algorithm is presented and explicit expressions for base 2 are given. To obtain an efficient component-by-component construction algorithm we exploit the structure of the underlying cyclic group. We compare our new higher order multivariate quadrature rules to existing quadrature rules based on higher order digital nets by computing their worst-case error. These numerical results show that the higher order polynomial lattice rules improve upon the known constructions of quasi-Monte Carlo rules based on higher order digital nets.