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Optimal Monte Carlo Methods for \(L^2\)-Approximation

  • David Krieg
Article
  • 67 Downloads

Abstract

We construct Monte Carlo methods for the \(L^2\)-approximation in Hilbert spaces of multivariate functions sampling not more than n function values of the target function. Their errors catch up with the rate of convergence and the preasymptotic behavior of the error of any algorithm sampling n pieces of arbitrary linear information, including function values.

Keywords

Approximation of multivariate functions Monte Carlo methods Optimal order of convergence Preasymptotic estimates Multivariate integration 

Mathematics Subject Classification

41A25 41A63 65C05 65D15 65D30 68Q25 65Y20 

Notes

Acknowledgements

I wish to thank Erich Novak, Robert Kunsch, Winfried Sickel, and two anonymous referees, whose comments and questions led to the present generality of the theorems.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität JenaJenaGermany

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