BIT Numerical Mathematics

, Volume 49, Issue 3, pp 543–564 | Cite as

On the power of standard information for L approximation in the randomized setting

  • Frances Y. Kuo
  • Grzegorz W. Wasilkowski
  • Henryk Woźniakowski
Article
  • 49 Downloads

Abstract

We study approximation of multivariate functions from a general separable reproducing kernel Hilbert space in the randomized setting with the error measured in the L norm. We consider algorithms that use standard information consisting of function values or general linear information consisting of arbitrary linear functionals. The power of standard or linear information is defined as, roughly speaking, the optimal rate of convergence of algorithms using n function values or linear functionals. We prove under certain assumptions that the power of standard information in the randomized setting is at least equal to the power of linear information in the worst case setting, and that the powers of linear and standard information in the randomized setting differ at most by 1/2. These assumptions are satisfied for spaces with weighted Korobov and Wiener reproducing kernels. For the Wiener case, the parameters in these assumptions are prohibitively large, and therefore we also present less restrictive assumptions and obtain other bounds on the power of standard information. Finally, we study tractability, which means that we want to guarantee that the errors depend at most polynomially on the number of variables and tend to zero polynomially in n−1 when n function values are used.

Keywords

Weighted multivariate approximation Randomized algorithms Monte Carlo methods Tractability 

Mathematics Subject Classification (2000)

41A63 65C05 65D15 65Y20 

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Copyright information

© Springer Science + Business Media B.V. 2009

Authors and Affiliations

  • Frances Y. Kuo
    • 1
  • Grzegorz W. Wasilkowski
    • 2
  • Henryk Woźniakowski
    • 3
    • 4
  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia
  2. 2.Department of Computer ScienceUniversity of KentuckyLexingtonUSA
  3. 3.Department of Computer ScienceColumbia UniversityNew YorkUSA
  4. 4.Institute of Applied MathematicsUniversity of WarsawWarszawaPoland

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