A Newton-type method for solving an inverse Sturm-Liouville problem
DOI: 10.1080/17415977.2014.947478
Title: A Newton-type method for solving an inverse Sturm-Liouville problem
Journal Title: Inverse Problems in Science and Engineering
Volume: Volume 23
Issue: Issue 5
Publication Date: July 2015
Start Page: 851
End Page: 883
Published online: 21 Aug 2014
ISSN: 1741-5977
Affiliations:
a Division of Physical and Computational Sciences, University of Pittsburgh at Bradford, Bradford, PA, USA.
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.
Accepted: 4 Jul 2014

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