Lattice rules for nonperiodic smooth integrands

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• 作者：Josef Dick (1)
  Dirk Nuyens (2)
  Friedrich Pillichshammer (3)
  
• 关键词：65D30 ; 65D32 ; 11K16 ; 11K36
• 刊名：Numerische Mathematik
• 出版年：2014
• 出版时间：February 2014
• 年：2014
• 卷：126
• 期：2
• 页码：259-291
• 全文大小：450 KB
• 作者单位：Josef Dick (1)
  Dirk Nuyens (2)
  Friedrich Pillichshammer (3)
  
  1. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia
  2. Department of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001, Heverlee, Belgium
  3. Institut f眉r Finanzmathematik, Universit盲t Linz, Altenbergerstr. 69, 4040, Linz, Austria
  
• ISSN：0945-3245

文摘

The aim of this paper is to show that one can achieve convergence rates of $N^{-\alpha + \delta }$ for $\alpha > 1/2$ (and for $\delta > 0$ arbitrarily small) for nonperiodic $\alpha $ -smooth cosine series using lattice rules without random shifting. The smoothness of the functions can be measured by the decay rate of the cosine coefficients. For a specific choice of the parameters the cosine series space coincides with the unanchored Sobolev space of smoothness 1. We study the embeddings of various reproducing kernel Hilbert spaces and numerical integration in the cosine series function space and show that by applying the so-called tent transformation to a lattice rule one can achieve the (almost) optimal rate of convergence of the integration error. The same holds true for symmetrized lattice rules for the tensor product of the direct sum of the Korobov space and cosine series space, but with a stronger dependence on the dimension in this case.