Skip to main content
SpringerLink
Log in

High dimensional polynomial interpolation on sparse grids

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

We study polynomial interpolation on a d-dimensional cube, where d is large. We suggest to use the least solution at sparse grids with the extrema of the Chebyshev polynomials. The polynomial exactness of this method is almost optimal. Our error bounds show that the method is universal, i.e., almost optimal for many different function spaces. We report on numerical experiments for d = 10 using up to 652 065 interpolation points.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. Berthold, W. Hoppe and B. Silbermann, A fast algorithm for solving the generalized airfoil equation, J. Comput. Appl. Math. 43 (1992) 185–219.

    Article  MATH  MathSciNet  Google Scholar 

  2. R. Cools, Integration over an hypercube, Unpublished manuscript (1998).

  3. C. de Boor and A. Ron, The least solution for the polynomial interpolation problem, Math. Z. 210 (1992) 347–378.

    MATH  MathSciNet  Google Scholar 

  4. C. de Boor and A. Ron, Computational aspects of polynomial interpolation in several variables, Math. Comp. 58 (1992) 705–727.

    Article  MATH  MathSciNet  Google Scholar 

  5. F.-J. Delvos, d-variate Boolean interpolation, J. Approx. Theory 34 (1982) 99–114.

    Article  MATH  MathSciNet  Google Scholar 

  6. V.K. Dzjadyk and V.V. Ivanov, On asymptotics and estimates for the uniform norms of the Lagrange interpolation polynomials corresponding to the Chebyshev nodal points, Anal. Math. 9 (1983) 85–97.

    Article  MATH  MathSciNet  Google Scholar 

  7. H. Ehlich and K. Zeller, Auswertung der Normen von Interpolationsoperatoren, Math. Ann. 164 (1966) 105–112.

    Article  MATH  MathSciNet  Google Scholar 

  8. K. Frank and S. Heinrich, Computing discrepancies of Smolyak quadrature rules, J. Complexity 12 (1996) 287–314.

    Article  MATH  MathSciNet  Google Scholar 

  9. A.C. Genz, Testing multidimensional integration routines, in: Tools, Methods, and Languages for Scientific and Engineering Computation, eds. B. Ford, J.C. Rault and F. Thomasset (North-Holland, Amsterdam, 1984) pp. 81–94.

  10. A.C. Genz, A package for testing multiple integration subroutines, in: Numerical Integration, eds. P. Keast and G. Fairweather (Kluwer, Dordrecht, 1987) pp. 337–340.

    Google Scholar 

  11. Th. Gerstner and M. Griebel, Numerical integration using sparse grids, Numer. Algorithms 18 (1998) 209–232.

    Article  MATH  MathSciNet  Google Scholar 

  12. G. Mastroianni and P. Vertesi, Weighted Lp error of Lagrange interpolation, J. Approx. Theory 82 (1995) 321–339.

    Article  MATH  MathSciNet  Google Scholar 

  13. E. Novak and K. Ritter, High dimensional integration of smooth functions over cubes, Numer. Math. 75 (1996) 79–97.

    Article  MATH  MathSciNet  Google Scholar 

  14. E. Novak and K. Ritter, The curse of dimension and a universal method for numerical integration, in: Multivariate Approximation and Splines, eds. G. Nürnberger, J. W. Schmidt and G. Walz, International Series of Numerical Mathematics, Vol. 125 (Birkhäuser, Basel, 1997) pp. 177–187.

    Google Scholar 

  15. E. Novak and K. Ritter, Simple cubature formulas with high polynomial exactness, Constr. Approx. 15 (1999) 499–522.

    Article  MATH  MathSciNet  Google Scholar 

  16. E. Novak, K. Ritter, R. Schmitt and A. Steinbauer, On a recent interpolatory method for high dimensional integration, J. Comput. Appl. Math. (1997, to appear).

  17. E. Novak, K. Ritter and A. Steinbauer, A multiscale method for the evaluation of Wiener integrals, in: Approximation Theory IX, Vol. 2, eds. C.K. Chui and L.L. Schumaker (Nashville, TN, 1998) pp. 251–258.

  18. J. Prestin, Lagrange interpolation on extended generalized Jacobi nodes, Numer. Algorithms 5 (1993) 179–190.

    Article  MATH  MathSciNet  Google Scholar 

  19. Th. Sauer, Polynomial interpolation of minimal degree, Numer. Math. 78 (1997) 59–85.

    Article  MATH  MathSciNet  Google Scholar 

  20. I.H. Sloan and S. Joe, Lattice Methods for Multiple Integration (Clarendon Press, Oxford, 1994).

    Google Scholar 

  21. S.A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Soviet Math. Dokl. 4 (1963) 240–243.

    Google Scholar 

  22. F. Sprengel, Interpolation and wavelets on sparse Gauss–Chebyshev grids, in: Multivariate Approximation, eds. W. Haussmann et al., Mathematical Research, Vol. 101 (Akademie Verlag, Berlin, 1997) pp. 269–286.

    Google Scholar 

  23. F. Sprengel, A unified approach to error estimates for interpolation on full and sparse Gauß– Chebyshev grids, Rostocker Math. Kolloq. 51 (1997) 51–64.

    MATH  MathSciNet  Google Scholar 

  24. V.N. Temlyakov, Approximation of Periodic Functions (Nova Science, New York, 1993).

    Google Scholar 

  25. G.W. Wasilkowski and H. Wózniakowski, Explicit cost bounds of algorithms for multivariate tensor product problems, J. Complexity 11 (1995) 1–56.

    Article  MATH  MathSciNet  Google Scholar 

  26. G.W.Wasilkowski and H. Wózniakowski, Weighted tensor-product algorithms for linear multivariate problems, J. Complexity 15 (1999) 402–447.

    Article  MATH  MathSciNet  Google Scholar 

  27. H. Wózniakowski, Tractability and strong tractability of linear multivariate problems, J. Complexity 10 (1994) 96–128.

    Article  MATH  MathSciNet  Google Scholar 

  28. H. Wózniakowski, Tractability and strong tractability of multivariate tensor product problems, J. Comput. Inform. 4 (1994) 1–19.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. 3SOFT, Wetterkreuz 19a, D-91058, Erlangen, Germany E-mail:

    Volker Barthelmann

  2. Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstraße 1 1/2, D-91054, Erlangen, Germany E-mail:

    Erich Novak

  3. Fakultät für Mathematik und Informatik, Universität Passau, D-94030, Passau, Germany E-mail:

    Klaus Ritter

Authors
  1. Volker Barthelmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Erich Novak
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Klaus Ritter
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barthelmann, V., Novak, E. & Ritter, K. High dimensional polynomial interpolation on sparse grids. Advances in Computational Mathematics 12, 273–288 (2000). https://doi.org/10.1023/A:1018977404843

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1018977404843

Search

Navigation