The Curse of Dimension and a Universal Method For Numerical Integration

  • Erich Novak
  • Klaus Ritter
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 125)

Abstract

Many high dimensional problems are difficult to solve for any numerical method. This curse of dimension means that the computational cost must increase exponentially with the dimension of the problem. A high dimension, however, can be compensated by a high degree of smoothness. We study numerical integration and prove that such a compensation is possible by a recently invented method. The method is shown to be universal, i.e., simultaneously optimal up to logarithmic factors, on two different smoothness scales. The first scale is defined by isotropic smoothness conditions, while the second scale involves anisotropic smoothness and is related to partially separable functions.

This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Babuška L, Über universal optimale Quadraturformeln, Apl. Mat. 13 (1968), 304–338 and 388-404.MATHGoogle Scholar
  2. 2.
    Baszenski G. and Delvos F.-J., Multivariate Boolean trapezoidal rules, in Approximation, Probability and Related Fields, Anastassiou G. (ed.), Plenum, New York, 1994, 109–117.CrossRefGoogle Scholar
  3. 3.
    Bellman R., Adaptive Control Processes: a Guided Tour, Princeton University, Princeton, 1961.MATHGoogle Scholar
  4. 4.
    Brass H., Universal quadrature rules in the space of periodic functions, in Numerical Integration III, Brass H. and Hämmerlin G. (eds.), ISNM 85 Birk-häuser, Basel, 1988, 16–24.Google Scholar
  5. 5.
    Delvos F.-J. and Schempp W., Boolean Methods in Interpolation and Approximation, Pitman Research Notes in Mathematics Series 230, Longman, Essex, 1989.MATHGoogle Scholar
  6. 6.
    Griebel M. and Oswald P., Tensor product type subspace splitting and multilevel iterative methods for anisotropic problems, Advances in Comp. Math. 4 (1995), 171–206.MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Griebel M., Schneider M., and Zenger Ch., A combination technique for the solution of sparse grid problems, in Iterative Methods in Linear Algebra, Beauwens R. and de Groen P. (eds.), Elsevier, North-Holland, 1992, 263–281.Google Scholar
  8. 8.
    Frank K. and Heinrich S., Computing discrepancies of Smolyak quadrature rules, J. Complexity 12 (1996), 287–314.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Frank K., Heinrich S., and Pereverzev S., Information complexity of multivariate Fredholm integral equations in Sobolev classes, J. Complexity 12 (1996), 17–34.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Müller-Gronbach Th., Hyperbolic cross designs for approximation of random fields, Preprint-Nr. 1811, Fachbereich Mathematik, TH Darmstadt, 1996.Google Scholar
  11. 11.
    Niederreiter H., Random Number Generation and Quasi-Monte Carlo Methods, SIAM, Philadelphia, 1992.CrossRefMATHGoogle Scholar
  12. 12.
    Novak E., Deterministic and Stochastic Error Bounds in Numerical Analysis, Lecture Notes in Mathematics 1349, Springer, Berlin, 1988.MATHGoogle Scholar
  13. 13.
    Novak E. and Ritter K., Global optimization using hyperbolic cross points, in State of the Art in Global Optimization, Floudas C. A. and Pardalos P. M. (eds.), Kluwer, Dordrecht, 1996, 19–33.CrossRefGoogle Scholar
  14. 14.
    Novak E. and Ritter K., High dimensional integration of smooth functions over cubes, Numer. Math. 75 (1996), 79–97.MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Novak E., Sloan I. H., and Wožniakowski H., Tractability of tensor product linear operators, preprint.Google Scholar
  16. 16.
    Paskov S. H. and Traub J. F., Faster valuation of financial derivatives, J. Portfolio Management 22 (1995), 113–120.CrossRefGoogle Scholar
  17. 17.
    Petras K., On the universality of the Gaussian quadrature formula, East J. Approx. 2 (1996), 427–438.MathSciNetMATHGoogle Scholar
  18. 18.
    Pöplau G. and Sprengel F., Some error estimates for periodic interpolation on full and sparse grids, in Proceedings of Chamonix 1996, Le Méhauté A., Rabut C., and Schumaker L. L. (eds.), Vanderbilt, Nashville, 1996, to appear.Google Scholar
  19. 19.
    Ritter K., Average Case Analysis of Numerical Problems, Habilitationsschrift, Erlangen, 1995.Google Scholar
  20. 20.
    Ritter K., Wasilkowski G. W. and Wožniakowski H., Multivariate integration and approximation for random fields satisfying Sacks-Ylvisaker conditions, Ann. Appl. Prob. 5 (1995), 518–540.CrossRefMATHGoogle Scholar
  21. 21.
    Sloan I. H. and Joe S., Lattice Methods for Multiple Integration, Clarendon, Oxford, 1994.MATHGoogle Scholar
  22. 22.
    Sloan I. H. and Smith W. E., Product-integration with the Clenshaw-Curtis and related points, Numer. Math. 30 (1978), 415–428.MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Smolyak S. A., Quadrature and interpolation formulas for tensor products of certain classes of functions, Soviet Math. Dokl. 4 (1963), 240–243.Google Scholar
  24. 24.
    Temlyakov V. N., On universal cubature formulas, Soviet Math. Dokl. 43 (1991), 39–42.MathSciNetMATHGoogle Scholar
  25. 25.
    Temlyakov V. N., On a way of obtaining lower estimates for the errors of quadrature formulas, Math. USSR Sbornik 71 (1992), 247–257.MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Temlyakov V. N., Approximation of Periodic Functions, Nova Science, New York, 1994.Google Scholar
  27. 27.
    Traub J. F., Wasilkowski G. W., and Wožniakowski H., Information-Based Complexity, Academic Press, San Diego, 1988.MATHGoogle Scholar
  28. 28.
    Vesely F. J., Computational Physics, an Introduction, Plenum, New York, 1994.MATHGoogle Scholar
  29. 29.
    Wasilkowski G. W. and Wožniakowski H., Explicit cost bounds of algorithms for multivariate tensor product problems, J. Complexity 11 (1995), 1–56.MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Wasilkowski G. W. and Wožniakowski H., On tractability of path integration, J. Math. Phys. 37 (1996), 2071–2088.MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Wožniakowski H., Information-based complexity, Ann. Rev. Comput. Sci. 1 (1986), 319–380.CrossRefGoogle Scholar
  32. 32.
    Wožniakowski H., Tractability and strong tractability of linear multivariate problems, J. Complexity 10 (1994), 96–128.MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Zenger Ch., Sparse grids, in Parallel Algorithms for Partial Differential Equations, Hackbusch W. (ed.), Vieweg, Braunschweig, 1991, 241–251.Google Scholar

Copyright information

© Springer Basel AG 1997

Authors and Affiliations

  • Erich Novak
    • 1
  • Klaus Ritter
    • 1
  1. 1.Mathematisches InstitutUniversität Erlangen-NürnbergErlangenGermany

Personalised recommendations