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Construction of Good Rank-1 Lattice Rules Based on the Weighted Star Discrepancy

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Monte Carlo and Quasi-Monte Carlo Methods 2004

Summary

The ‘goodness’ of a set of quadrature points in [0, 1]d may be measured by the weighted star discrepancy. If the weights for the weighted star discrepancy are summable, then we show that for n prime there exist n-point rank-1 lattice rules whose weighted star discrepancy is O(n−1+δ) for any δ>0, where the implied constant depends on δ and the weights, but is independent of d and n. Further, we show that the generating vector z for such lattice rules may be obtained using a component-by-component construction. The results given here for the weighted star discrepancy are used to derive corresponding results for a weighted Lp discrepancy.

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References

  1. J. Dick, G. Leobacher, F. Pillichshammer, Construction algorithms for digital nets with small weighted star discrepancy, SIAM J. Numer. Anal., to appear.

    Google Scholar 

  2. F.J. Hickernell, H. Niederreiter, The existence of good extensible rank-1 lattices, J. Complexity 19 (2003) 286–300.

    Article  MathSciNet  Google Scholar 

  3. E. Hlawka, Funktionen von beschränkter Variation in der Theorie der Gleichverteilung, Ann. Mat. Pura ed Appl. 54 (1961) 325–334.

    Article  MATH  MathSciNet  Google Scholar 

  4. S. Joe, Component by component construction of rank-1 lattice rules having O(n−1(ln(n))d) star discrepancy, in: H. Niederreiter (Ed.), Monte Carlo and Quasi-Monte Carlo Methods 2002, Springer, Berlin, 2004, pp. 293–298.

    Google Scholar 

  5. S. Joe, I.H. Sloan, On computing the lattice rule criterion R, Math. Comp. 59 (1992) 557–568.

    Article  MathSciNet  Google Scholar 

  6. N.M. Korobov, Properties and calculation of optimal coefficients, Soviet Math. Dokl. 1 (1960) 696–700.

    MATH  MathSciNet  Google Scholar 

  7. 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.

    Article  MATH  MathSciNet  Google Scholar 

  8. F.Y. Kuo, S. Joe, Component-by-component construction of good lattice rules with a composite number of points, J. Complexity 18 (2002) 943–976.

    Article  MathSciNet  Google Scholar 

  9. F.Y. Kuo, S. Joe, Component-by-component construction of good intermediaterank lattice rules, SIAM J. Numer. Anal. 41 (2003) 1465–1486.

    Article  MathSciNet  Google Scholar 

  10. H. Niederreiter, Existence of good lattice points in the sense of Hlawka, Monatsh. Math. 86 (1978) 203–219.

    Article  MATH  MathSciNet  Google Scholar 

  11. H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, SIAM, Philadelphia, 1992.

    Google Scholar 

  12. D. Nuyens, R. Cools, Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces, Math. Comp., to appear.

    Google Scholar 

  13. I.H. Sloan, S. Joe, Lattice Methods for Multiple Integration, Clarendon Press, Oxford, 1994.

    Google Scholar 

  14. I.H. Sloan, F.Y. Kuo, S. Joe, On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces, Math. Comp. 71 (2002) 1609–1640.

    Article  MathSciNet  Google Scholar 

  15. I.H. Sloan, F.Y. Kuo, S. Joe, Constructing randomly shifted lattice rules in weighted Sobolev spaces, SIAM J. Numer. Anal. 40 (2002) 1650–1665.

    Article  MathSciNet  Google Scholar 

  16. I.H. Sloan, A.V. Reztsov, Component-by-component construction of good lattice rules, Math. Comp. 71 (2002) 263–273.

    Article  MathSciNet  Google Scholar 

  17. I.H. Sloan, H. Woźniakowski, When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?, J. Complexity 14 (1998) 1–33.

    Article  MathSciNet  Google Scholar 

  18. X. Wang, I.H. Sloan, J. Dick, On Korobov lattice rules in weighted spaces, SIAM J. Numer. Anal. 42 (2004) 1760–1779.

    Article  MathSciNet  Google Scholar 

  19. S.K. Zaremba, Some applications of multidimensional integration by parts, Ann. Polon. Math. 21 (1968) 85–96.

    MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Waikato, Private Bag 3105, Hamilton, New Zealand

    Stephen Joe

Authors
  1. Stephen Joe
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Editor information

Editors and Affiliations

  1. Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore, 117543, Republic of Singapore

    Harald Niederreiter

  2. INRIA Sophia Antipolis, route des lucioles 2004, 06902, Sophia Antipolis Cedex, France

    Denis Talay

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© 2006 Springer-Verlag Berlin Heidelberg

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Joe, S. (2006). Construction of Good Rank-1 Lattice Rules Based on the Weighted Star Discrepancy. In: Niederreiter, H., Talay, D. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2004. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-31186-6_12

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