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The Strong Tractability of Multivariate Integration Using Lattice Rules

In: Monte Carlo and Quasi-Monte Carlo Methods 2002

Author

Listed:
  • Fred J. Hickernell

    (Hong Kong Baptist University and Peking University — HKBU Joint Research Institute for Applied Mathematics, Department of Mathematics)

  • Ian H. Sloan

    (University of New South Wales, School of Mathematics)

  • Grzegorz W. Wasilkowski

    (University of Kentucky, Department of Computer Science)

Abstract

Summary Although many applications involve integrals over unbounded domains, most of the theory for numerical approximation of integrals assumes that the integration domain is bounded. This article builds upon previous work by the authors that investigates the approximation of integrals over boxes that may be finite, semiinfinite or infinite in each coordinate direction. The integrand is sampled over a design, W -1(zi), that is a transformation of the nodeset of an integration lattice z i . The error bound for the numerical integration rule is shown to be a product of two terms: i) the discrepancy of the original design, z i , on the unit cube and ii) the variation of the integrand. Previously known convergence rates for extensible lattice rules on unit cubes are used to derive sufficient conditions for the strong tractability of integration over more general domains. The variation of the integrand depends on several factors, including the function W used to make the transformation of variables.

Suggested Citation

  • Fred J. Hickernell & Ian H. Sloan & Grzegorz W. Wasilkowski, 2004. "The Strong Tractability of Multivariate Integration Using Lattice Rules," Springer Books, in: Harald Niederreiter (ed.), Monte Carlo and Quasi-Monte Carlo Methods 2002, pages 259-273, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-18743-8_15
    DOI: 10.1007/978-3-642-18743-8_15
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