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Exponential convergence and tractability of multivariate integration for Korobov spaces


Authors: Josef Dick, Gerhard Larcher, Friedrich Pillichshammer and Henryk Woźniakowski
Journal: Math. Comp. 80 (2011), 905-930
MSC (2010): Primary 11K45, 65C05, 65D30
DOI: https://doi.org/10.1090/S0025-5718-2010-02433-0
Published electronically: November 2, 2010
MathSciNet review: 2772101
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Abstract: In this paper we study multivariate integration for a weighted Korobov space for which the Fourier coefficients of the functions decay exponentially fast. This implies that the functions of this space are infinitely times differentiable. Weights of the Korobov space monitor the influence of each variable and each group of variables. We show that there are numerical integration rules which achieve an exponential convergence of the worst-case integration error. We also investigate the dependence of the worst-case error on the number of variables $ s$, and show various tractability results under certain conditions on weights of the Korobov space. Tractability means that the dependence on $ s$ is never exponential, and sometimes the dependence on $ s$ is polynomial or there is no dependence on $ s$ at all.


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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

Gerhard Larcher
Affiliation: Institut für Finanzmathematik, Universität Linz, Altenbergstraße 69, A-4040 Linz, Austria
Email: gerhard.larcher@jku.at

Friedrich Pillichshammer
Affiliation: Institut für Finanzmathematik, Universität Linz, Altenbergstraße 69, A-4040 Linz, Austria
Email: friedrich.pillichshammer@jku.at

Henryk Woźniakowski
Affiliation: Department of Computer Science, Columbia University, New York, New York 10027, USA and Institute of Applied Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland
Email: henryk@cs.columbia.edu

DOI: https://doi.org/10.1090/S0025-5718-2010-02433-0
Keywords: Quasi-Monte Carlo, numerical integration, lattice rules, tractability.
Received by editor(s): July 13, 2009
Received by editor(s) in revised form: March 17, 2010
Published electronically: November 2, 2010
Additional Notes: The second author was supported by the Austrian Science Foundation (FWF), Project P21196
The third author was supported by the Austrian Research Foundation (FWF), Project S 9609, which is part of the Austrian Research Network “Analytic Combinatorics and Probabilistic Number Theory”.
The fourth author was partially supported by the National Science Foundation.
Article copyright: © Copyright 2010 American Mathematical Society

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