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Integration of Multivariate Haar Wavelet Series

  • Stefan Heinrich
  • Fred J. Hickernell
  • Rong-Xian Yue
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2251)

Abstract

This article considers the error of integrating multivariate Haar wavelet series by quasi-Monte Carlo rules using scrambled digital nets. Both the worst-case and random-case errors are analyzed. It is shown that scrambled net quadrature has optimal order. Moreover, there is a simple formula for the worst-case error.

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References

  1. 1.
    H. Faure, Discrépance de suites associées à un systéme de numération (en dimensions), Acta Arith. 41 (1982), 337–351.zbMATHMathSciNetGoogle Scholar
  2. 2.
    S. Heinrich, F. J. Hickernell, and R. X. Yue, Optimal quadrature for Haar wavelet spaces, 2001, submitted for publication to Math. Comp.Google Scholar
  3. 3.
    P. Hellekalek and G. Larcher (eds.), Random and quasi-random point sets, Lecture Notes in Statistics, vol. 138, Springer-Verlag, New York, 1998.zbMATHGoogle Scholar
  4. 4.
    F. J. Hickernell and H. S. Hong, The asymptotic efficiency of randomized nets for quadrature, Math. Comp. 68 (1999), 767–791.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    F. J. Hickernell and H. Woźniakowski, The price of pessimism for multidimensional quadrature, J. Complexity 17 (2001), to appear.Google Scholar
  6. 6.
    F. J. Hickernell and R. X. Yue, The mean square discrepancy of scrambled (t, s)-sequences, SIAM J. Numer. Anal. 38 (2001), 1089–1112.CrossRefMathSciNetGoogle Scholar
  7. 7.
    H. S. Hong and F. J. Hickernell, Implementing scrambled digital nets, 2001, submitted for publication to ACM TOMS.Google Scholar
  8. 8.
    G. Larcher, Digital point sets: Analysis and applications, In Hellekalek and Larcher [3], pp. 167–222.Google Scholar
  9. 9.
    -, On the distribution of digital sequences, Monte Carlo and quasi-Monte Carlo methods 1996 (H. Niederreiter, P. Hellekalek, G. Larcher, and P. Zinterhof, eds.), Lecture Notes in Statistics, vol. 127, Springer-Verlag, New York, 1998, pp. 109–123.Google Scholar
  10. 10.
    H. Niederreiter, Low discrepancy and low dispersion sequences, J. Number Theory 30 (1988), 51–70.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    -, Random number generation and quasi-Monte Carlo methods, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1992.Google Scholar
  12. 12.
    H. Niederreiter and C. Xing, Quasirandom points and global function fields, Finite Fields and Applications (S. Cohen and H. Niederreiter, eds.), London Math. Society Lecture Note Series, no. 233, Cambridge University Press, 1996, pp. 269–296.Google Scholar
  13. 13.
    -, Nets, (t, s)-sequences and algebraic geometry, In Hellekalek and Larcher [3], pp. 267–302.Google Scholar
  14. 14.
    A. B. Owen, Randomly permuted (t, m, s)-nets and (t, s)-sequences, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (H. Niederreiter and P. J.-S. Shiue, eds.), Lecture Notes in Statistics, vol. 106, Springer-Verlag, New York, 1995, pp. 299–317.Google Scholar
  15. 15.
    -, Monte Carlo variance of scrambled net quadrature, SIAM J. Numer. Anal. 34 (1997), 1884–1910.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    I. M. Sobol’, Multidimensional quadrature formulas and Haar functions (in Russian), Izdat. “Nauka”, Moscow, 1969.Google Scholar
  17. 17.
    R. X. Yue and F. J. Hickernell, The discrepancy of digital nets, 2001, submitted to J. Complexity.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Stefan Heinrich
    • 1
  • Fred J. Hickernell
    • 2
  • Rong-Xian Yue
    • 3
  1. 1.FB InformatikUniversität KaiserslauternKaiserslauternGermany
  2. 2.Department of MathematicsHong Kong Baptist UniversityKowloon TongHong Kong SAR, China
  3. 3.College of Mathematical ScienceShanghai Normal UniversityShanghaiChina

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