Advertisement

Quasi-Monte Carlo integration of digitally smooth functions by digital nets

  • Gerhard Larcher
  • Gottlieb Pirsic
  • Reinhard Wolf Salzburg
Conference paper
Part of the Lecture Notes in Statistics book series (LNS, volume 127)

Abstract

In a series of papers by the first author and several coauthors, a “digital lattice rule” for the numerical integration of digitally smooth functions by digital nets was developed and investigated. In this paper we give the general concepts of the method and we prove an error estimate which in some sense summarizes and generalizes the currently known error estimates in this field.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

5 References

  1. [1]
    P.L. Butzer and H.J. Wagner. Walsh—Fourier series and the concept of a derivative. Applicable Analysis, 3:29–46, 1973.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    G. Larcher, H. Niederreiter, and W. Ch. Schmid. Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration. Monatsh. Math., 121:231–253, 1996.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    G. Larcher and W. Ch. Schmid. Multivariate Walsh series, digital nets and quasi-Monte Carlo integration. In H. Niederreiter and P. J.-S. Shiue, editors, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (Las Vegas, 1994), volume 106 of Lecture Notes in Statistics, pages 252–262. Springer, New York, 1995.CrossRefGoogle Scholar
  4. [4]
    G. Larcher, W. Ch. Schmid, and R. Wolf. Representation of functions as Walsh series to different bases and an application to the numerical integration of high-dimensional Walsh series. Math. Comp., 63:701–716, 1994.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [5]
    G. Larcher, W. Ch. Schmid, and R. Wolf. Quasi-Monte Carlo methods for the numerical integration of multivariate Walsh series. Math. and Computer Modelling, 23(8/9):55–67, 1996.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    G. Larcher and C. Traunfellner. On the numerical integration of Walsh series by number-theoretic methods. Math. Comp., 63:277–291, 1994.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    H. Niederreiter. Point sets and sequences with small discrepancy. Monatsh. Math., 104:273–337, 1987.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    [ ] H. Niederreiter. Low-discrepancy point sets obtained by digital constructions over finite fields. Czechoslovak Math. J., 42:143–166, 1992.zbMATHMathSciNetGoogle Scholar
  9. [9]
    H. Niederreiter. Random Number Generation and Quasi-Monte Carlo Methods. Number 63 in CBMS—NSF Series in Applied Mathematics. SIAM, Philadelphia, 1992.CrossRefGoogle Scholar
  10. [10]
    H. Niederreiter and C. P. Xing. Low-discrepancy sequences and global function fields with many rational places. Finite Fields Appl., 2:241–273, 1996.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    H. Niederreiter and C. P. Xing. Quasirandom points and global function fields. In S. Cohen and H. Niederreiter, editors, Finite Fields and Applications (Glasgow, 1995), volume 233 of Lect. Note Series of the London Math. Soc., pages 269–296. Camb. Univ. Press, Cambridge, 1996.Google Scholar
  12. [12]
    C.W. Onneweer. Differentiability for Rademacher series on groups. Acta Sci. Math., 39:121–128, 1977.zbMATHMathSciNetGoogle Scholar
  13. [13]
    G. R. Pirsic. Schnell konvergierende Walshreihen über Gruppen. Master’s thesis, University of Salzburg, 1995.Google Scholar
  14. [14]
    R. Wolf. A distance measure on finite abelian groups and an application to quasi-Monte Carlo integration. Preprint, University of Salzburg, 1996.Google Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Gerhard Larcher
    • 1
  • Gottlieb Pirsic
    • 1
  • Reinhard Wolf Salzburg
    • 1
  1. 1.Institut für MathematikUniversität SalzburgSalzburgAustria

Personalised recommendations