Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

Reducing the construction cost of the component-by-component construction of good lattice rules


Authors: J. Dick and F. Y. Kuo
Journal: Math. Comp. 73 (2004), 1967-1988
MSC (2000): Primary 65D30, 65D32; Secondary 68Q25
Published electronically: August 19, 2003
MathSciNet review: 2059746
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Abstract: The construction of randomly shifted rank-$1$ lattice rules, where the number of points $n$ is a prime number, has recently been developed by Sloan, Kuo and Joe for integration of functions in weighted Sobolev spaces and was extended by Kuo and Joe and by Dick to composite numbers. To construct $d$-dimensional rules, the shifts were generated randomly and the generating vectors were constructed component-by-component at a cost of $O(n^2d^2)$ operations. Here we consider the situation where $n$ is the product of two distinct prime numbers $p$ and $q$. We still generate the shifts randomly but we modify the algorithm so that the cost of constructing the, now two, generating vectors component-by-component is only $O(n(p+q)d^2)$ operations. This reduction in cost allows, in practice, construction of rules with millions of points. The rules constructed again achieve a worst-case strong tractability error bound, with a rate of convergence $O(p^{-1+\delta}q^{-1/2})$ for $\delta>0$.


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  • 1. N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404. MR 0051437, 10.1090/S0002-9947-1950-0051437-7
  • 2. Dick, J. (2003). On the convergence rate of the component-by-component construction of good lattice rules, J. Complexity, submitted.
  • 3. Hardy, G. H., Littlewood, J. E. and Pólya, G. (1934). Inequalities, Cambridge University Press, Cambridge.
  • 4. Hickernell, F. J. and Hong, H. S. (2002). Quasi-Monte Carlo methods and their randomizations, Applied Probability (R. Chan, Y.-K. Kwok, D. Yao, and Q Zhang, eds.), AMS/IP Studies in Advanced Mathematics 26, American Mathematical Society, Providence, 59-77.
  • 5. Loo Keng Hua and Yuan Wang, Applications of number theory to numerical analysis, Springer-Verlag, Berlin-New York; Kexue Chubanshe (Science Press), Beijing, 1981. Translated from the Chinese. MR 617192
  • 6. N. M. Korobov, Properties and calculation of optimal coefficients, Soviet Math. Dokl. 1 (1960), 696–700. MR 0120768
  • 7. Kuo, F. Y. (2003). Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces, J. Complexity, 19, 301-320.
  • 8. Kuo, F. Y. and Joe, S. (2002). Component-by-component construction of good lattice rules with a composite number of points, J. Complexity, 18, 943-976.
  • 9. Sloan, I. H., Kuo, F. Y. and Joe, S. (2002). On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces, Math. Comp., 71, 1609-1640.
  • 10. Sloan, I. H., Kuo, F. Y. and Joe, S. (2002). Constructing randomly shifted lattice rules in weighted Sobolev spaces, SIAM J. Numer. Anal., 40, 1650-1665.
  • 11. I. H. Sloan and A. V. Reztsov, Component-by-component construction of good lattice rules, Math. Comp. 71 (2002), no. 237, 263–273. MR 1862999, 10.1090/S0025-5718-01-01342-4
  • 12. Ian H. Sloan and Henryk Woźniakowski, When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals?, J. Complexity 14 (1998), no. 1, 1–33. MR 1617765, 10.1006/jcom.1997.0463
  • 13. Sloan, I. H. and Wozniakowski, H. (2001). Tractability of multivariate integration for weighted Korobov classes, J. Complexity, 17, 697-721.

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Additional Information

J. Dick
Affiliation: School of Mathematics, The University of New South Wales, Sydney, New South Wales 2052, Australia
Email: josi@maths.unsw.edu.au

F. Y. Kuo
Affiliation: Department of Mathematics, The University of Waikato, Private Bag 3105, Hamilton, New Zealand
Address at time of publication: School of Mathematics, The University of New South Wales, Sydney, New South Wales 2052, Australia
Email: fkuo@maths.unsw.edu.au

DOI: http://dx.doi.org/10.1090/S0025-5718-03-01610-7
Keywords: Quasi--Monte Carlo, numerical integration, lattice rules
Received by editor(s): August 23, 2002
Received by editor(s) in revised form: February 16, 2003
Published electronically: August 19, 2003
Article copyright: © Copyright 2003 American Mathematical Society