A Newton-type method for solving an inverse Sturm-Liouville problem

Abstract

A Newton-type method is proposed for the recovery of the unknown coefficient function in the canonical Sturm-Liouville differential equation from two spectral data. Specifically, the two spectral data will be used to produce two Cauchy data, which in turn will serve as the input in a non-linear equation whose unknown is the coefficient function in the canonical Sturm-Liouville differential equation. This non-linear equation is to be solved numerically by the Newton method. Each Newton iterate requires that a Goursat-Cauchy boundary value problem be solved numerically. The Fréchet differentiability of the non-linear map is also discussed in the present paper. The numerical implementation of the Newton method for this inverse Sturm-Liouville problem is illustrated with examples, and it will be compared with a quasi-Newton method, and with a variational method discussed in previous literature. The numerical examples show that the Newton method needs fewer iterates to recover the true coefficient function than the quasi-Newton method. The Newton method is comparable with the variational method in terms of accuracy and number of iterates when the boundary parameters are given, and it requires a much smaller number of iterates than the variational method, when the boundary parameters are reconstructed from the available spectra.