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Digital inversive vectors can achieve polynomial tractability for the weighted star discrepancy and for multivariate integration


Authors: Josef Dick, Domingo Gomez-Perez, Friedrich Pillichshammer and Arne Winterhof
Journal: Proc. Amer. Math. Soc. 145 (2017), 3297-3310
MSC (2010): Primary 11K38, 11K45, 11T23, 65C05, 65C10
DOI: https://doi.org/10.1090/proc/13490
Published electronically: January 27, 2017
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Abstract: We study high-dimensional numerical integration in the worst-case setting. The subject of tractability is concerned with the dependence of the worst-case integration error on the dimension. Roughly speaking, an integration problem is tractable if the worst-case error does not grow exponentially fast with the dimension. Many classical problems are known to be intractable. However, sometimes tractability can be shown. Often such proofs are based on randomly selected integration nodes. Of course, in applications, true random numbers are not available and hence one mimics them with pseudorandom number generators. This motivates us to propose the use of pseudorandom vectors as underlying integration nodes in order to achieve tractability. In particular, we consider digital inverse vectors and present two examples of problems, the weighted star discrepancy and integration of Hölder continuous, absolute convergent Fourier and cosine series, where the proposed method is successful.


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Additional Information

Josef Dick
Affiliation: School of Mathematics and Statistics, The University of New South Wales, Sydney NSW 2052, Australia
Email: josef.dick@unsw.edu.au

Domingo Gomez-Perez
Affiliation: Faculty of Sciences, University of Cantabria, E-39071 Santander, Spain
Email: domingo.gomez@unican.es

Friedrich Pillichshammer
Affiliation: Department for Financial Mathematics and Applied Number Theory, Johannes Kepler University Linz, Altenbergerstr. 69, 4040 Linz, Austria
Email: friedrich.pillichshammer@jku.at

Arne Winterhof
Affiliation: Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstr. 69, 4040 Linz, Austria
Email: arne.winterhof@oeaw.ac.at

DOI: https://doi.org/10.1090/proc/13490
Keywords: Weighted star discrepancy, pseudorandom numbers, tractability, quasi-Monte Carlo
Received by editor(s): December 21, 2015
Received by editor(s) in revised form: September 9, 2016
Published electronically: January 27, 2017
Additional Notes: The research of the first author was supported under the Australian Research Councils Discovery Projects funding scheme (project number DP150101770). The research of the second author was supported by the Ministerio de Economia y Competitividad research project MTM2014-55421-P. The third and fourth authors were supported by the Austrian Science Fund (FWF): Projects F5509-N26 (third author) and F5511-N26 (fourth author), respectively, which are part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.
Dedicated: In memory of Joseph Frederick Traub (1932–2015) and Oscar Moreno de Ayala (1946–2015)
Communicated by: Walter Van Assche
Article copyright: © Copyright 2017 American Mathematical Society