Summary
The star discrepancy is a quantity for measuring the uniformity of a set of quadrature points and appears in the Koksma-Hlawka inequality. For integrals over [0, 1]d it is known that there exist d-dimensional rank-1 lattice rules having 0(n -1(ln(n))d) star discrepancy, where n is the number of points. Here we show that for n prime such rules may be obtained by constructing their generating vectors component by component. The rules are constructed to satisfy certain bounds on a quantity known as R. Bounds on the star discrepancy in terms of R then yield the desired O(n -1(In(n))d) star discrepancy.
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
Hickernell, F.J, Niederreiter, H.: The existence of good extensible rank-1 lattices. J. Complexity, 19, 286–300 (2003)
Hlawka, E.: Funktionen von beschränkter Variation in der Theorie der Gleichverteilung. Ann. Mat. Pura ed AppL, 54, 325–334 (1961)
Joe, S., Sloan, I.H.: On computing the lattice rule criterion R. Math. Comp., 59, 557–568 (1992)
Krommer, A.R., Ueberhuber, C.W.: Computational Integration. SIAM, Philadelphia (1998)
Niederreiter, H.: Existence of good lattice points in the sense of Hlawka. Monatsh. Math., 86, 203–219 (1978)
Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992)
Sloan, I.H., Joe, S.: Lattice Rules for Multiple Integration. Clarendon Press, Oxford (1994)
Sloan, I.H., Reztsov, A.V.: Component-by-component construction of good lattice rules. Math. Comp., 71, 263–273 (2002)
Zaremba, S.K.: Some applications of multidimensional integration by parts. Ann. Polon. Math., 21, 85–96 (1968)
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
Joe, S. (2004). Component by Component Construction of Rank-1 Lattice Rules HavingO(n -1(In(n))d) Star Discrepancy. 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_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-18743-8_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-20466-4
Online ISBN: 978-3-642-18743-8
eBook Packages: Springer Book Archive