Stochastic Approximation of Functions and Applications

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


We survey recent results on the approximation of functions from Sobolev spaces by stochastic linear algorithms based on function values. The error is measured in various Sobolev norms, including positive and negative degree of smoothness. We also prove some new, related results concerning integration over Lipschitz domains, integration with variable weights, and study tractability of generalized versions of indefinite integration and discrepancy.


  1. 1.
    R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.Google Scholar
  2. 2.
    P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.Google Scholar
  3. 3.
    R. M. Dudley, A course on empirical processes (École d’Été de Probabilités de Saint-Flour XII-1982). Lecture Notes in Mathematics 1097, 2–141, Springer-Verlag, New York, 1984.Google Scholar
  4. 4.
    S. Heinrich, Random approximation in numerical analysis, in: K. D. Bierstedt, A. Pietsch, W. M. Ruess, D. Vogt (Eds.), Functional Analysis, Marcel Dekker, New York, 1993, 123–171.Google Scholar
  5. 5.
    S. Heinrich, Randomized approximation of Sobolev embeddings, in: Monte Carlo and Quasi-Monte Carlo Methods 2006 (A. Keller, S. Heinrich, H. Niederreiter, eds.), Springer, Berlin, 2008, 445–459.Google Scholar
  6. 6.
    S. Heinrich, Randomized approximation of Sobolev embeddings II, J. Complexity 25 (2009), 455–472.Google Scholar
  7. 7.
    S. Heinrich, Randomized approximation of Sobolev embeddings III, J. Complexity 25 (2009), 473–507.Google Scholar
  8. 8.
    S. Heinrich, B. Milla, The randomized complexity of indefinite integration, J. Complexity 27 (2011), 352–382.Google Scholar
  9. 9.
    S. Heinrich, E. Novak, G. W. Wasilkowski, H. Woźniakowski, The inverse of the star-discrepancy depends linearly on the dimension, Acta Arithmetica 96 (2001), 279–302.Google Scholar
  10. 10.
    A. Hinrichs, Covering numbers, Vapnik-Červonenkis classes and bounds for the star-discrepancy, J. Complexity 20 (2004), 477–483.Google Scholar
  11. 11.
    H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer, Berlin-Heidelberg-New York, 1974.Google Scholar
  12. 12.
    M. Ledoux, M. Talagrand, Probability in Banach Spaces, Springer, Berlin-Heidelberg-New York, 1991.Google Scholar
  13. 13.
    P. Mathé, Random approximation of Sobolev embeddings, J. Complexity 7 (1991), 261–281.Google Scholar
  14. 14.
    E. Novak, Deterministic and Stochastic Error Bounds in Numerical Analysis, Lecture Notes in Mathematics 1349, Springer-Verlag, Berlin, 1988.Google Scholar
  15. 15.
    E. Novak, H. Triebel, Function spaces in Lipschitz domains and optimal rates of convergence for sampling, Constr. Approx. 23 (2006), 325–350.Google Scholar
  16. 16.
    E. Novak, H. Woźniakowski, Tractability of Multivariate Problems, Volume 1, Linear Information, European Math. Soc., Zürich, 2008.Google Scholar
  17. 17.
    E. Novak, H. Woźniakowski, Tractability of Multivariate Problems, Volume 2, Standard Information for Functionals, European Math. Soc., Zürich, 2010.Google Scholar
  18. 18.
    G. Pisier, Remarques sur les classes de Vapnik-Červonenkis, Ann. Inst. Henri Poincaré, Probab. Stat. 20 (1984), 287–298.Google Scholar
  19. 19.
    E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.Google Scholar
  20. 20.
    J. F. Traub, G. W. Wasilkowski, and H. Woźniakowski, Information-Based Complexity, Academic Press, 1988.Google Scholar
  21. 21.
    H. Triebel, Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration, European Math. Soc., Zürich, 2010.Google Scholar
  22. 22.
    J. Vybíral, Sampling numbers and function spaces, J. Complexity 23 (2007), 773–792.Google Scholar
  23. 23.
    G. W. Wasilkowski, Randomization for continuous problems, J. Complexity 5 (1989), 195–218.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of KaiserslauternKaiserslauternGermany

Personalised recommendations