Summary
We develop an algorithm for the construction of randomly shifted rank-1 lattice rules in d-dimensional weighted Sobolev spaces with a significantly reduced construction cost. The results shown here are an extension of earlier results by the present authors. In this new algorithm, the number of quadrature points n is a product of r distinct prime numbers p 1,…,p r. This allows us to reduce the construction cost to O(n(p 1 + … +p r)d 2), which represents a significant reduction, especially for large n. The constructed rules achieve a worst-case error bound with a rate of convergence of O(n(p 1 + δ p -1/22 ... p -1/2 r ) for any δ > 0. Numerical experiments were carried out for r = 2, 3, 4 and 5. The results demonstrate that it can be advantageous to choose n as a product of up to 5 primes.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Dick, J.: On the convergence rate of the component-by-component construction of good lattice rules. Submitted.
Dick, J., Kuo, F.Y.: Reducing the construction cost of the component-bycomponent construction of good lattice rules. Math. Comp., to appear.
Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1934)
Hickernell, F.J.: Lattice Rules: How well do they measure up? In: Hellekalek, P. and Larcher, G. (eds) Random and Quasi-Random Point Sets, Lecture Notes in Statistics, vol. 138. Springer-Verlag, New York, 109-166 (1998)
Hickernell, F.J., Niederreiter, H.: The existence of good extensible rank-1 lattices. J. Complexity, 19, 286–300 (2003)
Hickernell, F.J., Woźniakowski, H.: Integration and approximation in arbitrary dimensions. Adv. Comput. Math., 12, 25–58 (2000)
Hickernell, F.J., Woźniakowski, H.: Tractability of multivariate integration for periodic functions. J. Complexity, 17, 660–682 (2001)
Hua, L.K., Wang, Y.: Applications of Number Theory to Numerical Analysis. Springer Verlag, Berlin; Science Press, Beijing (1981)
Korobov, N.M.: Properties and calculation of optimal coefficients. Doklady Akademii Nauk SSSR, 132, 1009–10 (Russian). English transl.: Soviet Mathematics Doklady, 1, 696-700 (1960)
Sloan, I.H., Kuo, F.Y., Joe, S.: Constructing randomly shifted lattice rules in weighted Sobolev spaces. SIAM J. Numer. Anal., 40, 1650–1665 (2002)
Sloan, I.H., Reztsov, A.V.: Component-by-component construction of good lattice rules. Math. Comp., 71, 263–273 (2002)
Sloan, I.H., Woźniakowski, H.: When are quasi-Monte Carlo algorithms efficient for high dimensional integrals? J. Complexity, 14, 1–33 (1998)
Sloan, I.H., Woźniakowski, H.: Tractability of multivariate integration for weighted Korobov classes. J. Complexity, 17, 697–721 (2001)
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
Dick, J., Kuo, F.Y. (2004). Constructing Good Lattice Rules with Millions of Points. 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_10
Download citation
DOI: https://doi.org/10.1007/978-3-642-18743-8_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-20466-4
Online ISBN: 978-3-642-18743-8
eBook Packages: Springer Book Archive