Advertisement

Implementation of a Component-By-Component Algorithm to Generate Small Low-Discrepancy Samples

  • Benjamin DoerrEmail author
  • Michael Gnewuch
  • Magnus Wahlström
Conference paper

Abstract

In [B. Doerr, M. Gnewuch, P. Kritzer, F. Pillichshammer. Monte Carlo Methods Appl., 14:129–149, 2008], a component-by-component (CBC) approach to generate small low-discrepancy samples was proposed and analyzed. The method is based on randomized rounding satisfying hard constraints and its derandomization. In this paper we discuss how to implement the algorithm and present first numerical experiments. We observe that the generated points in many cases have a significantly better star discrepancy than what is guaranteed by the theoretical upper bound. Moreover, we exhibit that the actual discrepancy is mainly caused by the underlying grid structure, whereas the rounding errors have a negligible contribution. Hence to improve the algorithm, we propose and analyze a randomized point placement. We also study a hybrid approach which combines classical low-discrepancy sequences and the CBC algorithm.

Keywords

Monte Carlo Failure Probability Star Discrepancy Placement Error Cache Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Doerr, B., Gnewuch, M.: Construction of low-discrepancy point sets of small size by bracketing covers and dependent randomized rounding. In: A. Keller, S. Heinrich, H. Niederreiter (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2006, pp. 299–312. Springer-Verlag, Berlin Heidelberg (2008) CrossRefGoogle Scholar
  2. 2.
    Doerr, B., Gnewuch, M., Kritzer, P., Pillichshammer, F.: Component-by-component construction of low-discrepancy point sets of small size. Monte Carlo Methods Appl. 14, 129–149 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Doerr, B., Gnewuch, M., Srivastav, A.: Bounds and constructions for the star discrepancy via δ-covers. J. Complexity 21, 691–709 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Doerr, B., Wahlström, M.: Randomized rounding in the presence of a cardinality constraint. In: Proceedings of ALENEX, pp. 162–174. SIAM (2009) Google Scholar
  5. 5.
    Gnewuch, M.: On probabilistic results for the discrepancy of a hybrid-Monte Carlo sequence. J. Complexity 25, 312–317 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Gnewuch, M.: Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy. J. Complexity 24, 154–172 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Gnewuch, M., Srivastav, A., Winzen, C.: Finding optimal volume subintervals with k points and calculating the star discrepancy are NP-hard problems. J. Complexity 25, 115–127 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Halton, J.H.: On the efficiency of certain quasi-random sequences of points in evaluating multidimensional integrals. Numer. Math. 2, 84–90 (1960) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Heinrich, S.: Some open problems concerning the star-discrepancy. J. Complexity 19, 416–419 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Heinrich, S., Novak, E., Wasilkowski, G.W., Woźniakowski, H.: The inverse of the star-discrepancy depends linearly on the dimension. Acta Arith. 96, 279–302 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hickernell, F.J., Sloan, I.H., Wasilkowski, G.W.: On tractability of weighted integration over bounded and unbounded regions in ℝs. Math. Comp. 73, 1885–1901 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods, SIAM CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63. SIAM, Philadelphia (1992) zbMATHGoogle Scholar
  13. 13.
    Ökten, G., Tuffin, B., Burago, V.: A central limit theorem and improved error bounds for a hybrid-Monte Carlo sequence with applications in computational finance. J. Complexity 22, 435–458 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Raghavan, P.: Probabilistic construction of deterministic algorithms: Approximating packing integer programs. J. Comput. Syst. Sci. 37, 130–143 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Spanier, J.: Quasi-Monte Carlo methods for particle transport problems. In: H. Niederreiter, P.J.S. Shiue (eds.) Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, pp. 121–148. Springer-Verlag, Berlin (1995) Google Scholar
  16. 16.
    Spencer, J.: Ten Lectures on the Probabilistic Method. SIAM, Philadelphia (1987) zbMATHGoogle Scholar
  17. 17.
    Thiémard, E.: An algorithm to compute bounds for the star discrepancy. J. Complexity 17, 850–880 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Thiémard, E.: Optimal volume subintervals with k points and star discrepancy via integer programming. Math. Meth. Oper. Res. 54, 21–45 (2001). Extended version available at http://ina2.eivd.ch/Collaborateurs/etr/ zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Benjamin Doerr
    • 1
    Email author
  • Michael Gnewuch
  • Magnus Wahlström
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany

Personalised recommendations