Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abate J., Whitt W. The Fourier-series method for inverting transforms of probability distributions. Queueing Systems, 10: 5–88, 1992.
Bellman R., Kalaba R., Lockett J. Numerical Inversion of the Laplace Transform. Amer. Elsevier, New York, 1966.
Bohman H. A method to calculate the distribution function when the characteristic function is known. Ark. Mat., 4: 99–157, 1960.
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.
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.
Courant R., Hilbert D. Methoden der Mathematischen Physik, vol.1. J. Springer, Berlin, page 64, 1931.
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.
Dahlquist G. A “multigrid” extension of the FFT for the numerical inversion of Fourier and Laplace transforms. BIT, 33: 85–112, 1993.
Davies B., Martin B. Numerical inversion of the Laplace transform: a survey and comparison of methods. J. Comp. Phys., 33: 1–32, 1979.
De Boor C. A Practical Guide to Splines. Springer, New York, 1978.
Dubner H., Abate J. Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform. JACM, 15: 115–123, 1968.
Fettis H. E. Numerical calculation of certain definite integrals by Poisson’s summation formula. MTAC, 9: 85–92, 1955.
Gautschi W. On the condition of a matrix arising in the numerical inversion of the Laplace transform. Math. Comp., 23: 109–118, 1969.
Gautschi W. Attenuation factors in practical Fourier analysis. Numer. Math., 18: 373–400, 1972.
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.
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.
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.
Talbot A. The accurate numerical inversion of Laplace transforms. J. Inst. Maths. Applies., 23: 97–120, 1979.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to Walter Gautschi on the occasion of his 65th birthday
Rights and permissions
Copyright information
© 1994 Birkhäuser
About this chapter
Cite this chapter
Dahlquist, G. (1994). A Multigrid FFT Algorithm for Slowly Convergent Inverse Laplace Transforms. In: Zahar, R.V.M. (eds) Approximation and Computation: A Festschrift in Honor of Walter Gautschi. ISNM International Series of Numerical Mathematics, vol 119. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4684-7415-2_9
Download citation
DOI: https://doi.org/10.1007/978-1-4684-7415-2_9
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4684-7417-6
Online ISBN: 978-1-4684-7415-2
eBook Packages: Springer Book Archive