Weighted compound integration rules with higher order convergence for all N |
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Authors: | Fred J. Hickernell Peter Kritzer Frances Y. Kuo Dirk Nuyens |
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Affiliation: | 1. Department of Applied Mathematics, Illinois Institute of Technology, Room E1-208, 10 West 32nd Street, Chicago, IL, 60616, USA 2. Department of Financial Mathematics, University of Linz, Altenbergerstr. 69, 4040, Linz, Austria 3. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia 4. Department of Computer Science, K.U.Leuven, Celestijnenlaan 200A, 3001, Heverlee, Belgium
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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/?N1/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 − α for α > 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. |
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