# A Multigrid FFT Algorithm for Slowly Convergent Inverse Laplace Transforms

## 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.

## Keywords

Discrete Fourier Transform Coarse Grid Inverse Fourier Transform Spline Interpolation Laplace Transform## Preview

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