Numerical Algorithms

, Volume 46, Issue 4, pp 369–391 | Cite as

Periodization strategy may fail in high dimensions

  • Frances Y. Kuo
  • Ian H. Sloan
  • Henryk Woźniakowski
Original Paper

Abstract

We discuss periodization of smooth functions f of d variables for approximation of multivariate integrals. The benefit of periodization is that we may use lattice rules, which have recently seen significant progress. In particular, we know how to construct effectively a generator of the rank-1 lattice rule with n points whose worst case error enjoys a nearly optimal bound Cd,pn−p. Here Cd,p is independent of d or depends at most polynomially on d, and p can be arbitrarily close to the smoothness of functions belonging to a weighted Sobolev space with an appropriate condition on the weights. If F denotes the periodization for f then the error of the lattice rule for a periodized function F is bounded by Cd,pn−p∣∣F∣∣ with the norm of F given in the same Sobolev space. For small or moderate d, the norm of F is not much larger than the norm of f. This means that for small or moderate d, periodization is successful and allows us to use optimal properties of lattice rules also for non-periodic functions. The situation is quite different if d is large since the norm of F can be exponentially larger than the norm of f. This can already be seen for f = 1. Hence, the upper bound of the worst case error of the lattice rule for periodized functions is quite bad for large d. We conjecture not only that this upper bound is bad, but also that all lattice rules fail for large d. That is, if we fix the number of points n and let d go to infinity then the worst case error of any lattice rule is bounded from below by a positive constant independent of n. We present a number of cases suggesting that this conjecture is indeed true, but the most interesting case, when the sum of the weights of the corresponding Sobolev space is bounded in d, remains open.

Keywords

Periodization Multivariate integration Quasi-Monte Carlo methods Lattice rules Worst case error 

Mathematics Subject Classifications (2000)

65D30 65D32 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover Publications, New York (1964)MATHGoogle Scholar
  2. 2.
    Aronszajn, N.: Theory of reproducing kernels. Trans. Amer. Math. Soc. 68, 337–404 (1950)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Beckers, M., Haegemans, A.: Transformation of integrands using lattice rules. In: Espelid, T.O., Genz, A.C. (eds.) Numerical Integration: Recent Developments, Software and Applications, pp. 329–340. Kluwer Academic Publishers, Dordrecht (1992)Google Scholar
  4. 4.
    Dick, J.: On the convergence rate of the component-by-component construction of good lattice rules. J. Complexity 20, 493–522 (2004)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dick, J., Pillichshammer, F.: Strong tractability of multivariate integration of arbitrary high order using digitally shifted polynomial lattice rules. J. Complexity 23, 436–453 (2007)CrossRefGoogle Scholar
  6. 6.
    Dick, J., Sloan, I.H., Wang, X., Woźniakowski, H.: Liberating the weights. J. Complexity 20, 593–623 (2004)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dick, J., Sloan, I.H., Wang, X., Woźniakowski, H.: Good lattice rules in weighted Korobov spaces with general weights. Numer. Math. 103, 63–97 (2006)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hickernell, F.J.: Obtaining O(N  − 2 + ε) convergence for lattice quadrature rules. In: Fang, K.T., Hickernell, F.J., Niederreiter, H. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2000, pp. 274–289 Springer-Verlag, Berlin (2002)Google Scholar
  9. 9.
    Hickernell, F.J., Woźniakowski, H.: Tractability of multivariate integration for periodic functions. J. Complexity 17, 660–682 (2001)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Korobov, N.M.: Number-theoretic Methods in Approximate Analysis. Fizmatgiz, Moscow (1963)Google Scholar
  11. 11.
    Kuo, F.Y.: Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces. J. Complexity 19, 301–320 (2003)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kuo, F.Y., Sloan, I.H., Wasilkowski, G.W., Woźniakowski, H.: On decompositions of multivariate functions (in preparation)Google Scholar
  13. 13.
    Laurie, D.: Periodizing transformations for numerical integration J. Comput. Appl. Math. 66, 337–344 (1996)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Novak, E., Woźniakowski, H.: Intractability results for integration and discrepancy J. Complexity 17, 388–411 (2001)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Nuyens, D., Cools, R.: Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces. Math. Comp. 75, 903–920 (2006)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Sidi, A.: A new variable transformation for numerical integration. Int. Series Numer. Math. 112, 359–373 (1993)MathSciNetGoogle Scholar
  17. 17.
    Sidi, A.: Extension of a class of periodizing variable transformations for numerical integration. Math. Comp. 75, 327–343 (2006)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Sloan, I.H., Joe, S.: Lattice Methods for Multiple Integration. Oxford University Press, Oxford (1994)MATHGoogle Scholar
  19. 19.
    Sloan, I.H., Kuo, F.Y., Joe, S.: Constructing randomly shifted lattice rules in weighted Sobolev spaces. SIAM J. Numer. Anal. 40, 1650–1665 (2002)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Sloan, I.H., Reztsov, A.V.: Component-by-component construction of good lattice rules. Math. Comp. 71, 263–273 (2002)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Sloan, I.H., Wang, X., Woźniakowski, H.: Finite-order weights imply tractability of multivariate integration. J. Complexity 20, 46–74 (2004)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Sloan, I.H., Woźniakowski, H.: When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?. J. Complexity 14, 1–33 (1998)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Sloan, I.H., Woźniakowski, H.: Tractability of multivariate integration for weighted Korobov classes. J. Complexity 17, 697–721 (2001)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Sloan, I.H., Woźniakowski, H.: Tractability of integration in non-periodic and periodic weighted tensor product Hilbert spaces. J. Complexity 18, 479–499 (2002)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Wasilkowski, G.W., Woźniakowski, H.: Polynomial-time algorithms for multivariate linear problems with finite-order weights: worst case setting. Found. Comput. Math. 5, 451–491 (2005)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2007

Authors and Affiliations

  • Frances Y. Kuo
    • 1
  • Ian H. Sloan
    • 1
  • Henryk Woźniakowski
    • 2
    • 3
  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia
  2. 2.Department of Computer ScienceColumbia UniversityNew YorkUSA
  3. 3.Institute of Applied MathematicsUniversity of WarsawWarsawPoland

Personalised recommendations