Numerische Mathematik

, Volume 103, Issue 1, pp 63–97 | Cite as

Good Lattice Rules in Weighted Korobov Spaces with General Weights

  • Josef Dick
  • Ian H. Sloan
  • Xiaoqun Wang
  • Henryk Woźniakowski
Article

Abstract

We study the problem of multivariate integration and the construction of good lattice rules in weighted Korobov spaces with general weights. These spaces are not necessarily tensor products of spaces of univariate functions. Sufficient conditions for tractability and strong tractability of multivariate integration in such weighted function spaces are found. These conditions are also necessary if the weights are such that the reproducing kernel of the weighted Korobov space is pointwise non-negative. The existence of a lattice rule which achieves the nearly optimal convergence order is proven. A component-by-component (CBC) algorithm that constructs good lattice rules is presented. The resulting lattice rules achieve tractability or strong tractability error bounds and achieve nearly optimal convergence order for suitably decaying weights. We also study special weights such as finite-order and order-dependent weights. For these special weights, the cost of the CBC algorithm is polynomial. Numerical computations show that the lattice rules constructed by the CBC algorithm give much smaller worst-case errors than the mean worst-case errors over all quasi-Monte Carlo rules or over all lattice rules, and generally smaller worst-case errors than the best Korobov lattice rules in dimensions up to hundreds. Numerical results are provided to illustrate the efficiency of CBC lattice rules and Korobov lattice rules (with suitably chosen weights), in particular for high-dimensional finance problems.

Keywords

Quasi-Monte Carlo methods lattice rules multivariate integration 

Mathematics Subject Classification (2000)

65C05 65D30 65D32 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Josef Dick
    • 1
  • Ian H. Sloan
    • 1
  • Xiaoqun Wang
    • 1
    • 2
  • Henryk Woźniakowski
    • 3
    • 4
  1. 1.School of MathematicsUniversity of New South WalesSydneyAustralia
  2. 2.Department of Mathematical SciencesTsinghua UniversityBeijingChina
  3. 3.Department of Computer ScienceColumbia UniversityNew YorkUSA
  4. 4.Institute of Applied Mathematics and MechanicsUniversity of WarsawPoland

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