An Intermediate Bound on the Star Discrepancy

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 23)

Abstract

Let \({P}_{n}(z)\) denote the point set of an n-point rank-1 lattice rule with generating vector \(z\). A criterion used to assess the ‘goodness’ of the point set \({P}_{n}(z)\) is the star discrepancy, \({D}^{{_\ast}}({P}_{n}(z))\). As calculating the star discrepancy is an NP-hard problem, then it is usual to work with bounds on it. In particular, it is known that the following two bounds hold:
$${D}^{{_\ast}}({P}_{ n}(z)) \leq 1 - {(1 - 1/n)}^{d} + T(z,n) \leq 1 - {(1 - 1/n)}^{d} + R(z,n)/2,$$
where d is the dimension and the quantities \(T(z,n)\) and \(R(z,n)\) are defined in the paper. Here we provide an intermediate bound on the star discrepancy by introducing a new quantity \(W(z,n)\) which satisfies
$$T(z,n) \leq W(z,n) \leq R(z,n)/2.$$
Like \(R(z,n)\), the quantity \(W(z,n)\) may be calculated to a fixed precision in O(nd) operations. A component-by-component construction based on \(W(z,n)\) is analysed. We present the results of numerical calculations which indicate that values of \(W(z,n)\) are much closer to \(T(z,n)\) than to \(R(z,n)/2\).

Notes

Acknowledgements

This work was carried out when the author was a Visiting Fellow in the School of Mathematics and Statistics at the University of New South Wales. The author acknowledges the hospitality received and gives particular thanks to Dr Frances Kuo and Professor Ian Sloan. The author also thanks Dr Vasile Sinescu for his useful comments on an earlier version of this paper.

References

  1. 1.
    Bundschuh, P., Zhu, Y.: A method for exact calculation of the discrepancy of low-dimensional finite point sets I. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 63, 115–133 (1993)Google Scholar
  2. 2.
    Dick, J., Pillichshammer, F.: Digital Nets and Sequences. Cambridge University Press, New York (2010)Google Scholar
  3. 3.
    Gnewuch, M., Srivastav, A., Winzen, C.: Finding optimal subintervals with k points and calculating the star discrepancy are NP-hard problems. Journal of Complexity 25, 115–127 (2009)Google Scholar
  4. 4.
    Gradshteyn, I.S., Ryzhik, I.M., Jeffrey, A., Zwillinger, D.: Tables of Integrals, Series and Products (7th edition). Academic Press, San Diego (2007)Google Scholar
  5. 5.
    Hlawka, E.: Funktionen von beschränkter Variation in der Theorie der Gleichverteilung. Annali di Matematica Pura ed Applicata 54, 325–333 (1961)Google Scholar
  6. 6.
    Joe, S.: Component by component construction of rank-1 lattice rules having \(O({n}^{-1}{(\ln (n))}^{d})\) star discrepancy. In: Niederreiter, H. (ed.) Monte Carlo and Quasi-Monte Carlo Methods 2002, pp. 293–298. Springer, Berlin (2004)Google Scholar
  7. 7.
    Joe, S., Sloan, I.H.: On computing the lattice rule criterion R. Mathematics of Computation 59, 557–568 (1992)Google Scholar
  8. 8.
    Niederreiter, H.: Existence of good lattice points in the sense of Hlawka. Monatshefte für Mathematik 86, 203–219 (1978)Google Scholar
  9. 9.
    Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992)Google Scholar
  10. 10.
    Sinescu, V., L’Ecuyer, P.: On the behavior of the weighted star discrepancy bounds for shifted lattice rules. In: L’Ecuyer, P., Owen, A.B. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2008, pp. 603–616. Springer, Berlin (2009)Google Scholar
  11. 11.
    Zaremba, S.K.: Some applications of multidimensional integration by parts. Annales Polonici Mathematici 21, 85–96 (1968)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsThe University of WaikatoHamiltonNew Zealand

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