A Multigrid FFT Algorithm for Slowly Convergent Inverse Laplace Transforms

  • Germund Dahlquist
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 119)


The algorithm is based on an exact relation, due to Cooley, Lewis and Welch, between the Discrete Fourier Transform and the periodic sums, associated with a function and its Fourier Transform in a similar way as in the Poisson summation formula. It makes use of several equidistant grids, with the same number of points covering m different symmetric intervals of length L, 2L, 4L, 8L,…, where it applies FFT and spline interpolation to the midpoints of the grid.

Typically the number of arithmetic operations per computed function value is about twice as large as for the FFT, but the distribution of points is more adequate for many applications, because the union of the grids is, globally, an approximately equidistant grid on a logarithmic scale. Some properties and applications of the algorithm will be discussed.

Novel features emphasized in this article include a new initial condition, where there is some relation to the work of Walter Gautschi on Practical Fourier Analysis, and the use of convolution smoothing to improve the performance on problems with discontinuities in the time domain.


Discrete Fourier Transform Coarse Grid Inverse Fourier Transform Spline Interpolation Laplace Transform 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Abate J., Whitt W. The Fourier-series method for inverting transforms of probability distributions. Queueing Systems, 10: 5–88, 1992.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Bellman R., Kalaba R., Lockett J. Numerical Inversion of the Laplace Transform. Amer. Elsevier, New York, 1966.MATHGoogle Scholar
  3. [3]
    Bohman H. A method to calculate the distribution function when the characteristic function is known. Ark. Mat., 4: 99–157, 1960.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Cooley J., Lewis P., Welch P. Application of the Fast Fourier Transform to computation of Fourier integrals, Fourier series, and convolution integrals. IEEE Trans., AU-15: 79–84, 1967.Google Scholar
  5. [5]
    Cooley J., Lewis P., Welch P. The Fast Fourier Transform: programming considerations in the calculation of sine, cosine and Laplace transforms. J. Sound and Vibration, 12: 315–337, 1970.MATHCrossRefGoogle Scholar
  6. [6]
    Courant R., Hilbert D. Methoden der Mathematischen Physik, vol.1. J. Springer, Berlin, page 64, 1931.CrossRefGoogle Scholar
  7. [7]
    Dahlquist G. On an inversion formula for Laplace transforms that uses the real part only. Report TRTITA-NA-9213, NADA, Royal Inst. Techn., Stockholm, 1992.Google Scholar
  8. [8]
    Dahlquist G. A “multigrid” extension of the FFT for the numerical inversion of Fourier and Laplace transforms. BIT, 33: 85–112, 1993.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Davies B., Martin B. Numerical inversion of the Laplace transform: a survey and comparison of methods. J. Comp. Phys., 33: 1–32, 1979.MathSciNetMATHCrossRefGoogle Scholar
  10. 10]
    De Boor C. A Practical Guide to Splines. Springer, New York, 1978.MATHCrossRefGoogle Scholar
  11. 11]
    Dubner H., Abate J. Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform. JACM, 15: 115–123, 1968.MathSciNetMATHCrossRefGoogle Scholar
  12. 12]
    Fettis H. E. Numerical calculation of certain definite integrals by Poisson’s summation formula. MTAC, 9: 85–92, 1955.MathSciNetMATHGoogle Scholar
  13. 13]
    Gautschi W. On the condition of a matrix arising in the numerical inversion of the Laplace transform. Math. Comp., 23: 109–118, 1969.MathSciNetMATHCrossRefGoogle Scholar
  14. 14]
    Gautschi W. Attenuation factors in practical Fourier analysis. Numer. Math., 18: 373–400, 1972.MathSciNetMATHCrossRefGoogle Scholar
  15. 15]
    Gustafson S-A Computing inverse Laplace transforms using convergence acceleration. In Proc. of the Second Conference on Computing and Control, Aug. 1990. Birkhauser, Zurich, 1990. K. L. Bowers and J. Lund, eds.Google Scholar
  16. 16]
    Hodgkinson D. F., Lever D. A., England T. H. Mathematical modelling of radionuclide migration through fractured rock using numerical inversion of Laplace transforms. Ann. Nucl. Energy, 11: 111, 1984.CrossRefGoogle Scholar
  17. 17]
    Pizarro M., Eriksson R. Modelling of the ground mode of transmission lines in time domain simulations. In 7th Int. Symp. on High Voltage Engineering, Aug. 1991, Dresden.Google Scholar
  18. 18]
    Talbot A. The accurate numerical inversion of Laplace transforms. J. Inst. Maths. Applies., 23: 97–120, 1979.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser 1994

Authors and Affiliations

  • Germund Dahlquist
    • 1
  1. 1.Department of Computing SciencesRoyal Institute of TechnologyStockholmSweden

Personalised recommendations