Weighted compound integration rules with higher order convergence for all N

  • Authors:

  • Fred J. Hickernell;Peter Kritzer;Frances Y. Kuo;Dirk Nuyens

  • Affiliations:

  • Department of Applied Mathematics, Illinois Institute of Technology, Chicago, USA 60616;Department of Financial Mathematics, University of Linz, Linz, Austria 4040;School of Mathematics and Statistics, University of New South Wales, Sydney, Australia 2052;Department of Computer Science, K.U.Leuven, Heverlee, Belgium 3001

  • Venue:
  • Numerical Algorithms
  • Year:
  • 2012

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Abstract

Quasi-Monte Carlo integration rules, which are equal-weight sample averages of function values, have been popular for approximating multivariate integrals due to their superior convergence rate of order close to 1/N or better, compared to the order $1/\sqrt{N}$ of simple Monte Carlo algorithms. For practical applications, it is desirable to be able to increase the total number of sampling points N one or several at a time until a desired accuracy is met, while keeping all existing evaluations. We show that although a convergence rate of order close to 1/N can be achieved for all values of N (e.g., by using a good lattice sequence), it is impossible to get better than order 1/N convergence for all values of N by adding equally-weighted sampling points in this manner. We then prove that a convergence of order N 驴驴驴驴 for 驴驴驴1 can be achieved by weighting the sampling points, that is, by using a weighted compound integration rule. We apply our theory to lattice sequences and present some numerical results. The same theory also applies to digital sequences.