Summary
Although many applications involve integrals over unbounded domains, most of the theory for numerical approximation of integrals assumes that the integration domain is bounded. This article builds upon previous work by the authors that investigates the approximation of integrals over boxes that may be finite, semiinfinite or infinite in each coordinate direction. The integrand is sampled over a design, W -1(zi), that is a transformation of the nodeset of an integration lattice z i . The error bound for the numerical integration rule is shown to be a product of two terms: i) the discrepancy of the original design, z i , on the unit cube and ii) the variation of the integrand. Previously known convergence rates for extensible lattice rules on unit cubes are used to derive sufficient conditions for the strong tractability of integration over more general domains. The variation of the integrand depends on several factors, including the function W used to make the transformation of variables.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
F. J. Hickernell and H. S. Hong, Computing multivariate normal probabilities using rank-1 lattice sequences, Proceedings of the Workshop on Scientific Computing (Hong Kong) (G. H. Golub, S. H. Lui, F. T. Luk, and R. J. Plemmons, eds.), Springer-Verlag, Singapore, 1997, pp. 209-215.
F. J. Hickernell, H. S. Hong, P. L’Éc uyer, and C. Lemieux, Extensible lattice sequences for quasi-Monte Carlo quadrature, SIAM J. Sci. Comput. 22 (2000), 1117–1138.
F. J. Hickernell, A generalized discrepancy and quadrature error bound, Math.Comp. 67 (1998), 299–322.
F. J. Hickernell and H. Niederreiter, The existence of good extensible rank-1 lattices, J. Complexity 19 (2003), 286–300.
F. J. Hickernell, I. H. Sloan, and G. W. Wasilkowski, On strong tractability of weighted multivariate integration, Math. Comp. (2004), to appear.
I. H. Sloan, and G. W. Wasilkowski, On tractability of weighted integration for certain Banach spaces of functions, Monte Carlo and Quasi-Monte Carlo Methods 2002 (H. Niederreiter, ed.), Springer-Verlag, Berlin, 2004, to appear.
I. H. Sloan, and G. W. Wasilkowski, On tractability of weighted integration over bounded and unbounded regions in ℝ s, Math. Comp. (2004), to appear.
L. K. Hua and Y. Wang, Applications of number theory to numerical analysis, Springer-Verlag and Science Press, Berlin and Beijing, 1981.
F.Y. Kuo and S. Joe, Component-by-component construction of good lattice rules with a composite number of points, J. Complexity 18 (2002), 943–976.
F. Y. Kuo, Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces, J. Complexity 19 (2003), 301–320.
H. Niederreiter, Random number generation and quasi-Monte Carlo methods, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1992.
I. H. Sloan and S. Joe, Lattice methods for multiple integration, Oxford University Press, Oxford, 1994.
I. H. Sloan, F. Y. Kuo, and S. Joe, Constructing randomly shifted lattice rules in weighted Sobolev spaces, SIAM J. Numer. Anal. 40 (2002), 1650–1665. [SKJ02b]_F. Y. Kuo, and S. Joe, On the step-by-step construction of quasi-Monte Carlo integation rules that achieve strong tractability error bounds in weighted Sobolev spaces, Math. Comp. 71 (2002), 1609–1640.
I. H. Sloan and A. V. Reztsov, Component-by-component construction of good lattice rules, Math. Comp. 71 (2002), 263–273.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hickernell, F.J., Sloan, I.H., Wasilkowski, G.W. (2004). The Strong Tractability of Multivariate Integration Using Lattice Rules. In: Niederreiter, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2002. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18743-8_15
Download citation
DOI: https://doi.org/10.1007/978-3-642-18743-8_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-20466-4
Online ISBN: 978-3-642-18743-8
eBook Packages: Springer Book Archive