Constructive Approximation

, 30:495 | Cite as

QMC Rules of Arbitrary High Order: Reproducing Kernel Hilbert Space Approach

Article

Abstract

In this paper we consider numerical integration of smooth functions lying in a particular reproducing kernel Hilbert space. We show that the worst-case error of numerical integration in this space converges at the optimal rate, up to some power of a log N factor. A similar result is shown for the mean square worst-case error, where the bound for the latter is always better than the bound for the square worst-case error. Finally, bounds for integration errors of functions lying in the reproducing kernel Hilbert space are given. The paper concludes by illustrating the theory with numerical results.

Keywords

Quasi-Monte Carlo Numerical integration Digital nets Reproducing kernel Hilbert spaces 

Mathematics Subject Classification (2000)

65D30 65D32 11K38 11K45 

References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1971) Google Scholar
  2. 2.
    Caflisch, R.E., Morokoff, W.J., Owen, A.B.: Valuation of mortgage backed securities using Brownian Bridge to reduce effective dimension. J. Comput. Finance 1, 27–46 (1997) Google Scholar
  3. 3.
    Chrestenson, H.E.: A class of generalized Walsh functions. Pac. J. Math. 5, 17–31 (1955) MATHMathSciNetGoogle Scholar
  4. 4.
    Dick, J.: Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high-dimensional periodic functions. SIAM J. Numer. Anal. 45, 2141–2176 (2007) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dick, J.: The decay of the Walsh coefficients of smooth functions. Bull. Aust. Math. Soc. (2009, to appear) Google Scholar
  6. 6.
    Dick, J.: Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal. 46, 1519–1553 (2008) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dick, J., Baldeaux, J.: Equidistribution properties of generalized nets and sequences. In: L’Ecuyer, P., Owen. A.B. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2008. Springer (2010, to appear) Google Scholar
  8. 8.
    Dick, J., Kritzer, P.: Duality theory and propagation rules for digital nets of higher order. Math. Comput. (2010, to appear) Google Scholar
  9. 9.
    Dick, J., Niederreiter, H.: On the exact t-value of Niederreiter and Sobol’ sequences. J. Complex. 24, 572–581 (2008) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Dick, J., Pillichshammer, F.: Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces. J. Complex. 21, 149–195 (2005) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dick, J., Pillichshammer, F.: On the mean square weighted ℒ2 discrepancy of randomized digital (t,m,s)-nets over ℤ2. Acta Arith. 117, 371–403 (2005) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Dick, J., Kuo, F., Pillichshammer, F., Sloan, I.H.: Construction algorithms for polynomial lattice rules for multivariate integration. Math. Comput. 74, 1895–1921 (2005) MathSciNetGoogle Scholar
  13. 13.
    Dick, J., Kritzer, P., Pillichshammer, F., Schmid, W.Ch.: On the existence of higher order polynomial lattices based on a generalized figure of merit. J. Complex. 23, 581–593 (2007) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Faure, H.: Discrépance de suites associées à un système de numération (en dimension s). Acta Arith. 41, 337–351 (1982) MATHMathSciNetGoogle Scholar
  15. 15.
    Gerstner, T., Griebel, M.: Numerical integration using sparse grids. Numer. Algorithms 18, 209–232 (1998) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    L’Ecuyer, P., Lemieux, C.: Variance reduction via lattice rules. Manag. Sci. 46, 1214–1235 (2000) CrossRefGoogle Scholar
  17. 17.
    L’Ecuyer, P., Lemieux, C.: Recent advances in randomized quasi-Monte Carlo methods. In: Dror, M., L’Ecuyer, P., Szidarovszki, F. (eds.) Modeling Uncertainty. Internat. Ser. Oper. Res. Management Sci., vol. 46, pp. 419–474. Kluwer Academic, Boston (2002) Google Scholar
  18. 18.
    Matoušek, J.: Geometric Discrepancy. Algorithms Combin., vol. 18. Springer, Berlin (1999) MATHGoogle Scholar
  19. 19.
    Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1992) MATHGoogle Scholar
  20. 20.
    Niederreiter, H., Pirsic, G.: Duality for digital nets and its applications. Acta Arith. 97, 173–182 (2001) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Niederreiter, H., Xing, Ch.: Global function fields with many rational places and their applications. In: Mullin, R.C., Mullen, G.L. (eds.) Finite Fields: Theory, Applications, and Algorithms, Waterloo, ON, 1997. Contemp. Math., vol. 225, pp. 87–111. Amer. Math. Soc., Providence (1999) Google Scholar
  22. 22.
    Pirsic, G.: A software implementation of Niederreiter–Xing sequences. In: Fang, K.-T., Hickernell, F.J., Niederreiter, H. (eds.) Monte Carlo and Quasi-Monte Carlo Methods, 2000 (Hong Kong), pp. 434–445. Springer, Berlin (2002) Google Scholar
  23. 23.
    Sharygin, I.F.: A lower estimate for the error of quadrature formulas for certain classes of functions. Zh. Vychisl. Mat. i Mat. Fiz. 3, 370–376 (1963) Google Scholar
  24. 24.
    Sloan, I.H., Joe, S.: Lattice Methods for Multiple Integration. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1994) MATHGoogle Scholar
  25. 25.
    Sloan, I.H., Woźniakowski, H.: When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals? J. Complex. 14, 1–33 (1998) MATHCrossRefGoogle Scholar
  26. 26.
    Sloan, I.H., Woźniakowski, H.: Tractability of multivariate integration for weighted Korobov classes. J. Complex. 17, 697–721 (2001) MATHCrossRefGoogle Scholar
  27. 27.
    Smolyak, S.A.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR 4, 240–243 (1963) Google Scholar
  28. 28.
    Sobol’, I.M.: Distribution of points in a cube and approximate evaluation of integrals. Zh. Vychisl. Mat. i Mat. Fiz. 7, 784–802 (1967) MathSciNetGoogle Scholar
  29. 29.
    Walsh, J.L.: A closed set of normal orthogonal functions. Am. J. Math. 45, 5–24 (1923) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia

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