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BIT Numerical Mathematics

, Volume 37, Issue 4, pp 846–861 | Cite as

Quasi-monte carlo methods for numerical integration of multivariate Haar series

  • Karl Entacher
Article

Abstract

In the present paper we study quasi-Monte Carlo methods to integrate functions representable by generalized Haar series in high dimensions. Using (t, m, s)-nets to calculate the quasi-Monte Carlo approximation, we get best possible estimates of the integration error for practically relevant classes of functions. The local structure of the Haar functions yields interesting new aspects in proofs and results. The results are supplemented by concrete computer calculations.

AMS subject classification

Primary 65D30 42C10 

Key words

Numerical integration generalized Haar functions low-discrepancy point sets quasi-Monte Carlo methods 

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References

  1. 1.
    K. Entacher,Generalized Haar function systems in the theory of uniform distributions of sequences modulo one, PhD thesis, University of Salzburg, 1995.Google Scholar
  2. 2.
    K. Entacher,Generalized Haar function systems, digital nets and quasi-Monte Carlo integration, in Wavelet Applications III, Proc. SPIE 2762, H. H. Szu, ed., 1996. Available on the internet athttp://random.mat.sbg.ac.at. Google Scholar
  3. 3.
    P. Hellekalek,General discrepancy estimates III: The Erdös-Turàn-Koksma inequality for the Haar function system, Monatsh. Math., 120 (1995), pp. 25–45.MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    N. M. Korobov,Number-Theoretic Methods in Approximate Analysis, Fizmatgiz, Moscow, 1963. (In Russian).Google Scholar
  5. 5.
    G. Larcher, A. Lauß, H. Niederreiter, and W. Ch. Schmid,Optimal polynomials for (t, m, s)-nets and numerical integration of Walsh series, SIAM J. Numer. Analysis, 33 (1996), pp. 2239–2253.MATHCrossRefGoogle Scholar
  6. 6.
    G. Larcher and W. Ch. Schmid,Multivariate Walsh series, digital nets and quasi-Monte Carlo integration, in Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, H. Niederreiter and P. Jau-Shyong Shiue, eds., volume 106, Lecture Notes in Statistics, Springer, pp. 252–262, 1995.Google Scholar
  7. 7.
    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 (1994), pp. 701–716.MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    G. Larcher, W. Ch. Schmid, and R. Wolf,Quasi-Monte Carlo methods for the numerical integration of multivariate Walsh series, Math. Comput. Modelling, 23:8/9 (1996), pp. 55–67.MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    G. Larcher and C. Traunfellner,On the numerical integration of Walsh series by number-theoretic methods, Math. Comp., 63 (1994), pp. 277–291.MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    H. Niederreiter,Point sets and sequences with small discrepancy, Monatsh. Math., 104 (1987), pp. 273–337.MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    H. Niederreiter,Low-discrepancy and low-dispersion sequences, J. Number Theory, 30 (1988), pp. 51–70.MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    H. Niederrwiter,Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia, USA, 1992.Google Scholar
  13. 13.
    I. M. Sobol',The distribution of points in a cube and the approximate evaluation of integrals, Zh. Vycisl. Mat. i Mat Fiz., 7 (1967), pp. 784–802. (In Russian).MATHMathSciNetGoogle Scholar
  14. 14.
    I. M. Sobol',Multidimensional Quadrature Formulas and Haar Functions, Nauka., Moscow, 1969. (In Russian).Google Scholar

Copyright information

© Swets & Zeitlinger 1997

Authors and Affiliations

  • Karl Entacher
    • 1
  1. 1.Department of MathematicsUniversity of SalzburgSalzburgAustria

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