Numerical determination of potentials
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In this paper we will describe some computational methods for the recovery of potentials under a variety of types of data measurements. We will look at both inverse Sturm-Liouville problems on a finite interval and inverse scattering problems on the line. The unifying approach to all of this is the fact that many of these types of problems can be solved by converting the given spectral or scattering data into boundary data for a certain hyperbolic partial differential equation. In all cases the problem is an overdetermined one and it is precisely this fact that allows us to recover the potential. Further, we will show that this translated problem can be solved in a numerically stable way, and indeed this approach leads to an excellent, as well as a unifying, scheme for the reconstruction of the potential. There is a classical method of solving many of the above types of problems and the definitive formulation is due to Gel'fand, Levitan and Marchenko some forty years ago. Our approach has many similarities, and indeed the same starting point, but the crucial difference is while the original scheme reduced the problem to a Fredholm integral equation, we will exploit instead an equivalent hyperbolic partial differential equation.
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