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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Exponential convergence and tractability of multivariate integration for Korobov spaces
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by Josef Dick, Gerhard Larcher, Friedrich Pillichshammer and Henryk Woźniakowski PDF
Math. Comp. 80 (2011), 905-930 Request permission

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
  • MR Author ID: 661956
  • ORCID: 0000-0001-6952-9218
  • 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
  • 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.
  • © Copyright 2010 American Mathematical Society
  • Journal: Math. Comp. 80 (2011), 905-930
  • MSC (2010): Primary 11K45, 65C05, 65D30
  • DOI: https://doi.org/10.1090/S0025-5718-2010-02433-0
  • MathSciNet review: 2772101