On the step-by-step construction of quasi: Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces

  • Authors:

  • I. H. Sloan;F. Y. Kuo;S. Joe

  • Affiliations:

  • School of Mathematics, University of New South Wales, Sydney, New South Wales 2052, Australia;Department of Mathematics, University of Waikato, Private Bag 3105, Hamilton, New Zealand;Department of Mathematics, University of Waikato, Private Bag 3105, Hamilton, New Zealand

  • Venue:
  • Mathematics of Computation
  • Year:
  • 2002

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Abstract

We develop and justify an algorithm for the construction of quasi-Monte Carlo (QMC) rules for integration in weighted Sobolev spaces; the rules so constructed are shifted rank-1 lattice rules. The parameters characterising the shifted lattice rule are found "component-by-component": the (d+1)-th component of the generator vector and the shift are obtained by successive 1-dimensional searches, with the previous d components kept unchanged. The rules constructed in this way are shown to achieve a strong tractability error bound in weighted Sobolev spaces. A search for n-point rules with n prime and all dimensions 1 to d requires a total cost of O(n3d2) operations. This may be reduced to O(n3d) operations at the expense of O(n2) storage. Numerical values of parameters and worst-ease errors are given for dimensions up to 40 and n up to a few thousand. The worst-case errors for these rules are found to be much smaller than the theoretical bounds.