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Monte Carlo and Quasi-Monte Carlo Methods 1996 pp 238–250Cite as

Computing Discrepancies Related to Spaces of Smooth Periodic Functions

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Part of the book series: Lecture Notes in Statistics ((LNS,volume 127))

Abstract

A notion of discrepancy is introduced, which represents the integration error on spaces of r-smooth periodic functions. It generalizes the diaphony and constitutes a periodic counterpart to the classical L2-discrepancy as well as r-smooth versions of it introduced recently by Paskov [Pas93]. Based on previous work [FH96], we develop an efficient algorithm for computing periodic discrepancies for quadrature formulas possessing certain tensor product structures, in particular, for Smolyak quadrature rules (also called sparse grid methods). Furthermore, fast algorithms of computing periodic discrepancies for lattice rules can easily be derived from well—known properties of lattices. On this basis we carry out numerical comparisons of discrepancies between Smolyak and lattice rules.

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Author information

Authors and Affiliations

  1. RUS, Universität Stuttgart, Allmandring 30a, Stuttgart, D-70550, Germany

    Karin Frank

  2. FB Informatik, Universität Kaiserslautern, PF 3049, Kaiserslautern, D-67653, Germany

    Stefan Heinrich

Authors
  1. Karin Frank
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  2. Stefan Heinrich
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Editor information

Editors and Affiliations

  1. Institute of Information Processing, Austrian Academy of Sciences, Vienna, A-1010, Austria

    Harald Niederreiter

  2. Department of Mathematics, University of Salzburg, Salzburg, A-5020, Austria

    Peter Hellekalek , Gerhard Larcher  & Peter Zinterhof ,  & 

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© 1998 Springer Science+Business Media New York

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Frank, K., Heinrich, S. (1998). Computing Discrepancies Related to Spaces of Smooth Periodic Functions. In: Niederreiter, H., Hellekalek, P., Larcher, G., Zinterhof, P. (eds) Monte Carlo and Quasi-Monte Carlo Methods 1996. Lecture Notes in Statistics, vol 127. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1690-2_15

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  • DOI: https://doi.org/10.1007/978-1-4612-1690-2_15

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-98335-6

  • Online ISBN: 978-1-4612-1690-2

  • eBook Packages: Springer Book Archive

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