Skip to main content
SpringerLink
Log in

Fast NFFT Based Summation of Radial Functions

  • Published:
Sampling Theory in Signal and Image Processing Aims and scope Submit manuscript

Abstract

This paper is concerned with the fast summation of radial functions by the fast Fourier transform for nonequispaced data. We enhance the fast summation algorithm proposed in [20] by introducing a new regularization procedure based on the two-point Taylor interpolation by algebraic polynomials and estimate the corresponding approximation error. Our error estimates are more sophisticated than those in [20]. Beyond the kernels Kß(x) = 1/|x|ß(ß ∈ N), we are also interested in the generalized multiquadrics which play an important role in the approximation of functions by-radial basis functions.

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. R. P. Agarwal and P. J. Y. Wong. Error inequalities in polynomial interpolation and their applications, volume 262 of Mathematics and its Applications. Kluwer Academic Publishers, 1993.

  2. R. K. Beatson and W. A. Light. Fast evaluation of radial basis functions: methods for two-dimensional polyharmonic splines. IMA J. Numer. Anal, 17:343–372, 1997.

    Article  MathSciNet  Google Scholar 

  3. R. K. Beatson and G. N. Newsam. Fast evaluation of radial basis functions. I. Comput. Math. Appl, 24:7–19, 1992.

    Article  MathSciNet  Google Scholar 

  4. G. Beylkin. On the fast Fourier transform of functions with singularities. Appl. Comput. Harmon. Anal, 2(4):363–381, 1995.

    Article  MathSciNet  Google Scholar 

  5. J. B. Cherrie, R. K. Beatson, and G. N. Newsam. Fast evaluation of radial basis functions: methods for generalized multiquadrics in Rn. SIAM J. Sci. Comput., 23:1549–1571, 2002.

    Article  MathSciNet  Google Scholar 

  6. C. de Boor. Total positivity of the spline collocation matrix. Indiana Univ. Math. J., 25(6):541–551, 1976.

    Article  MathSciNet  Google Scholar 

  7. C. de Boor. Splinefunktionen. Lectures in Mathematics ETH Zurich. Birkhäuser Verlag, Basel, 1990. With a preface by M. Trummer.

    Google Scholar 

  8. A. Dutt and V. Rokhlin. Fast Fourier transforms for nonequispaced data. SIAM J. Sci. Comput, 14:1368–1393, 1993.

    Article  MathSciNet  Google Scholar 

  9. J. A. Fessler and B. P. Sutton, NUFFT–nonuniform FFT toolbox for Matlab. http://www.eecs.umich.edu/bpsutton/MR/Code/NUFFT/=,2002.

  10. R. Franke. Scattered data interpolation: tests of some methods. Math. Comp., 38(157):181–200, 1982.

    MathSciNet  MATH  Google Scholar 

  11. L. Greengard. The rapid evaluation of potential fields in particle systems. MIT Press, Cambridge, MA, 1988.

    Book  Google Scholar 

  12. L. Greengard and V. Rokhlin. A fast algorithm for particle simulations. J. Comput. Phys., 73:325–318. 1987.

    Article  MathSciNet  Google Scholar 

  13. W. Hackbusch. A sparse matrix arithmetic based on H-matrices. I. Introduction to H-matrices. Computing, 62:89–108, 1999.

    Article  MathSciNet  Google Scholar 

  14. W. Hackbusch and B. N. Khoromskij. A sparse H-matrix arithmetic. II. Application to multi-dimensional problems. Computing, 64:21–17. 2000.

    MathSciNet  MATH  Google Scholar 

  15. W. Hackbusch and Z. P. Nowak. On the fast matrix multiplication in the boundary element method by panel clustering. Numer. Math., 54:463–491, 1989.

    Article  MathSciNet  Google Scholar 

  16. S. Kunis and D. Potts. NFFT, software package, C subroutine library. http://www.math.uni-luebeck.de/potts/nfft, 2002.

    Google Scholar 

  17. S. Kunis, D. Potts, and G. Steidl. Fast Fourier transform at nonequispaced knots - a user’s guide to a C-library. Preprint, MU-Lübeck B-02-13, September 2002.

    Google Scholar 

  18. C. Müller. Über die ganzen Lösungen der Wellengleichung. Math. Annalen, 124:235–261. 1952.

    Article  MathSciNet  Google Scholar 

  19. A. Nieslony, D. Potts, and G. Steidl. Rapid evaluation of radial functions by fast Fourier transforms at nonequispaced knots. Preprint, MU-Lübeck A02-11, May 2002.

    MATH  Google Scholar 

  20. D. Potts and G. Steidl. Fast summation at nonequispaced knots by NFFTs. SIAM J. Set. Comput, to appear.

  21. D. Potts, G. Steidl, and M. Tasclie. Fast Fourier transforms for nonequispaced data: a tutorial. In Modern sampling theory, Appl. Num. Harm. Anal., pages 217–270. Birkhäuser, Boston, MA, 2001.

    Google Scholar 

  22. M. J. D. Powell. The theory of radial basis function approximation in 1990. In Advances in numerical analysis, Vol. II (Lancaster, 1990), Oxford Sci. Publ., pages 105–210. Oxford Univ. Press, New York, 1992.

    MATH  Google Scholar 

  23. E. E. Tyrtyshnikov. Mosaic-skeleton approximations. Calcolo, 33:17–57. 1996.

    MathSciNet  MATH  Google Scholar 

  24. E. E. Tyrtyshnikov. Mosaic ranks and skeletons. In L. G. Vulkov, J. Wasniewski, and P. Y. Yalamov, editors, Numerical Analysis and its Applications (Rousse, 1996), volume 1196 of Lecture Notes in Computer Science, pages 505–516. Springer, Berlin, 1997.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Mathematics and Computer Science, University of Mannheim, D-68131, Mannheim, Germany

    M. Fenn & G. Steidl

Authors
  1. M. Fenn
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Steidl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Fenn.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fenn, M., Steidl, G. Fast NFFT Based Summation of Radial Functions. STSIP 3, 5–28 (2004). https://doi.org/10.1007/BF03549403

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03549403

Key words and phrases

2000 AMS Mathematics Subject Classification

Search

Navigation