Strong tractability of multivariate integration of arbitrary high order using digitally shifted polynomial lattice rules

Authors:
Josef Dick;Friedrich Pillichshammer
Affiliations:
University of New South Wales Asia, 1 Kay Siang Road, Singapore 248922, Singapore;Institut für Finanzmathematik, Universität Linz, Altenbergerstraße 69, A-4040 Linz, Austria
Venue:
Journal of Complexity
Year:
2007

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Abstract

In this paper we prove the existence of digitally shifted polynomial lattice rules which achieve strong tractability results for Sobolev spaces of arbitrary high smoothness. The convergence rate is shown to be the best possible up to a given degree of smoothness of the integrand. Indeed we even show the existence of polynomial lattice rules which automatically adjust themselves to the smoothness of the integrand up to a certain given degree. Further we show that strong tractability under certain conditions on the weights can be obtained and that polynomial lattice rules exist for which the worst-case error can be bounded independently of the dimension. These results hold independent of the smoothness.