THE NUMERICAL-SOLUTION OF INVERSE PROBLEMS OF FOURIER CONVOLUTION TYPE

A novel method for the solution of inverse problems posed as first-kind Fredholm integral equations of Fourier convolution type is developed. Such problems arise in many areas of physical modelling, and two examples are discussed herein. Conventional Fourier transform numerical techniques cannot be directly applied to the type of problem given in this paper, since the data function and the kernel function do not possess classical Fourier transforms. The key element in this new approach is to recast the integral equation to yield a data function that does possess a classical transform. Several alternative methods of achieving this are discussed. The new method has the advantage of dramatically exhibiting the effects of noise on a successful recovery of the object function. The exponential error growth in the frequency domain can be utilized to accurately limit the data to the low-frequency regime in which signal dominates noise. The recovery problem is then reduced to a high-frequency extrapolation problem in Fourier space. This paper illustrates the efficiency of the Biraud extrapolation technique in problems of this type.