Advertisement

Scrambled Polynomial Lattice Rules for Infinite-Dimensional Integration

  • Jan Baldeaux
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 23)

Abstract

In the random case setting, scrambled polynomial lattice rules, as discussed in Baldeaux and Dick (Numer. Math. 119:271–297, 2011), enjoy more favorable strong tractability properties than scrambled digital nets. This short note discusses the application of scrambled polynomial lattice rules to infinite-dimensional integration. In Hickernell et al. (J Complex 26:229–254, 2010), infinite-dimensional integration in the random case setting was examined in detail, and results based on scrambled digital nets were presented. Exploiting these improved strong tractability properties of scrambled polynomial lattice rules and making use of the analysis presented in Hickernell et al. (J Complex 26:229–254, 2010), we improve on the results that were achieved using scrambled digital nets.

Keywords

Quadrature Rule Reproduce Kernel Hilbert Space Sampling Regime Lattice Rule Integration Node 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Baldeaux, J., Dick, J., A construction of polynomial lattice rules with small gain coefficients, Numerische Mathematik, 119, 271–297, 2011.Google Scholar
  2. 2.
    Creutzig, J., Dereich, S., Müller-Gronbach, T., Ritter, K., Infinite-dimensional quadrature and approximation of distributions, Foundations of Computational Mathematics, 9, 391–429, 2009.Google Scholar
  3. 3.
    Heinrich, S., Monte Carlo complexity of global solution of integral equations, Journal of Complexity, 14, 151–175, 1998.Google Scholar
  4. 4.
    Hickernell, F.J., Müller-Gronbach, T., Niu, B., Ritter, K., Multi-level Monte Carlo Algorithms for Infinite-Dimensional Integration on \({\mathbb{R}}^{\mathbb{N}}\), Journal of Complexity, 26, 229–254, 2010.Google Scholar
  5. 5.
    Giles, M.B., Multilevel Monte Carlo path simulation, Operations Research, 56, 607–617, 2008.Google Scholar
  6. 6.
    Gnewuch, M., Infinite-dimensional Integration on Weighted Hilbert Spaces, Mathematics of Computation, 2012.Google Scholar
  7. 7.
    Gnewuch, M., Weighted geometric discrepancies and numerical integration on reproducing kernel Hilbert spaces, Journal of Complexity 28, 2–17, 2012.Google Scholar
  8. 8.
    Imai, J., Kawai, R., Quasi-Monte Carlo Method for Infinitely Divisible Random Vectors via Series Representations, SIAM Journal on Scientific Computing, 32, 1879–1897, 2010.Google Scholar
  9. 9.
    Kuo, F.Y., Sloan, I.H., Wasilkowski, G.W., Woźniakowski, H., Liberating the dimension, Journal of Complexity, 26, 422–454, 2010.Google Scholar
  10. 10.
    Niu, B., Hickernell, F.J., Monte Carlo simulation of stochastic integrals when the cost of function evaluation is dimension dependent, Monte Carlo and Quasi-Monte Carlo Methods 2008 (P. L’Ecuyer and A.Owen, eds.), Springer-Verlag, Berlin, 545–560, 2010.Google Scholar
  11. 11.
    Niu, B., Hickernell, F.J., Müller-Gronbach, T., Ritter, K., Deterministic Multi-level Algorithms for Infinite-Dimensional Integration on \({\mathbb{R}}^{\mathbb{N}}\), Journal of Complexity, 26, 229–254, 2010.Google Scholar
  12. 12.
    Novak, E., Deterministic and stochastic error bounds in numerical analysis, Lecture Notes in Mathematics, 1349, Springer-Verlag, Berlin, 1988.Google Scholar
  13. 13.
    Plaskota, L., Wasilkowski, G.W., Tractability of infinite-dimensional integration in the worst case and randomized settings, Journal of Complexity, 27, 505–518, 2011.Google Scholar
  14. 14.
    Ritter, K., Average-case analysis of numerical problems, Lecture Notes in Mathematics, 1733, Springer-Verlag, Berlin, 2000.Google Scholar
  15. 15.
    Traub, J., Wasilkowski, G.W., Woźniakowski, H., Information-based Complexity, Academic Press, New York, 1988.Google Scholar
  16. 16.
    Yue, R.-X., Hickernell, F.J., Strong tractability of integration using scrambled Niederreiter points, Mathematics of Computation, 74, 1871–1893, 2005.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.School of Finance and EconomicsUniversity of TechnologySydneyAustralia

Personalised recommendations