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

Article
  • 24 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.

References

  1. 1.
    Babenko, K.I.: About the approximation of periodic functions of many variable trigonometric polynomials. Dokl. Akad. Nauk SSR 32, 247–250 (1960)Google Scholar
  2. 2.
    Cohen, A., Davenport, M.A., Leviatan, D.: On the stability and accuracy of least squares approximations. Found. Comput. Math. 13, 819–834 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cohen, A., Migliorati, G.: Optimal weighted least-squares methods. SMAI J. Comput. Math. 3, 181–203 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Heinrich, S.: Multilevel Monte Carlo methods. In: Proceedings of the Third International Conference on Large-Scale Scientific Computing, Sozopol (Bulgaria), pp. 58–67. Springer, Berlin (2001)Google Scholar
  5. 5.
    Heinrich, S.: Random approximation in numerical analysis. In: Proceedings of the Conference Functional Analysis, Essen (Germany), pp. 123–171. Marcel Dekker (1994)Google Scholar
  6. 6.
    Heinrich, S.: Randomized approximation of Sobolev embeddings. In: Keller, A., Heinrich, S., Niederreiter, H. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2006, pp. 445–459. Springer, Berlin (2008)CrossRefGoogle Scholar
  7. 7.
    Hinrichs, A., Novak, E., Vybíral, J.: Linear information versus function evaluations for \(L_2\)-approximation. J. Approx. Theory 153, 97–107 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Jerome, J.W.: On the \(L_2\) n-width of certain classes of functions of several variables. J. Math. Anal. Appl. 20, 110–123 (1967)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Krieg, D.: Tensor power sequences and the approximation of tensor product operators. J. Complex. 44, 30–51 (2018)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Krieg, D., Novak, E.: A universal algorithm for multivariate integration. Found. Comput. Math. 17(4), 895–916 (2017)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kühn, T., Mayer, S., Ullrich, T.: Counting via entropy: new preasymptotics for the approximation numbers of Sobolev embeddings. SIAM J. Numer. Anal. 54(6), 3625–3647 (2016)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kühn, T., Sickel, W., Ullrich, T.: Approximation of mixed order Sobolev functions on the \(d\)-torus—asymptotics, preasymptotics and \(d\)-dependence. Constr. Approx. 42, 353–398 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Mathé, P.: Random approximation of Sobolev embeddings. J. Complex. 7, 261–281 (1991)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Mityagin, B.S.: Approximation of functions in \(L^p\) and \(C\) on the torus. Math. Notes 58, 397–414 (1962)MATHGoogle Scholar
  15. 15.
    Nikol’skaya, N.S.: Approximation of differentiable functions of several variables by Fourier sums in the \(L_p\)-metric. Sibirsk. Mat. Zh. 15, 395–412 (1974); English transl. in Siberian Math. J. 15 (1974)Google Scholar
  16. 16.
    Novak, E.: Optimal linear randomized methods for linear operators in Hilbert spaces. J. Complex. 8, 22–36 (1992)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems. Volume I: Linear Information. EMS, Zürich (2008)CrossRefMATHGoogle Scholar
  18. 18.
    Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems. Volume III: Standard Information for Operators. EMS, Zürich (2012)CrossRefMATHGoogle Scholar
  19. 19.
    Sickel, W., Ullrich, T.: Spline interpolation on sparse grids. Appl. Anal. 90, 337–383 (2010)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Traub, J.F., Wasilkowski, G.W., Woźniakowski, H.: Information-Based Complexity. Academic Press, Cambridge (1988)MATHGoogle Scholar
  21. 21.
    Triebel, H.: Sampling numbers and embedding constants. Proc. Steklov Inst. Math. 248, 268–277 (2005)MathSciNetMATHGoogle Scholar
  22. 22.
    Ullrich, M.: A Monte Carlo method for integration of multivariate smooth functions. SIAM J. Numer. Anal. 55(3), 1188–1200 (2017)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Wasilkowski, G.W., Woźniakowski, H.: The power of standard information for multivariate approximation in the randomized setting. Math. Comput. 76, 965–988 (2006)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität JenaJenaGermany

Personalised recommendations