Journal of Integral Equations and Applications

A nonlinear integral equation and an iterative algorithm for an inverse source problem

Rainer Kress and William Rundell

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We consider the inverse problem of recovering the shape of an extended source of known homogeneous strength within a conducting medium from one voltage and current measurement on the accessible boundary of the medium and present an iterative solution method via boundary integral equations. The main idea of our approach is to equivalently reformulate the inverse source problem as an inverse boundary value problem with a non-local Robin condition on the boundary of the source domain. Following our approach in \cite{KR} for an inverse Dirichlet problem, from Green's representation formula we obtain a nonlinear integral equation for the unknown boundary curve which can be solved by regularized Newton iterations. We present the foundations of the inverse algorithm and illustrate its feasibility by some numerical examples.

Article information

J. Integral Equations Applications, Volume 27, Number 2 (2015), 179-197.

First available in Project Euclid: 9 September 2015

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Zentralblatt MATH identifier

Primary: 31A25: Boundary value and inverse problems 45Q05: Inverse problems 49N45: Inverse problems


Kress, Rainer; Rundell, William. A nonlinear integral equation and an iterative algorithm for an inverse source problem. J. Integral Equations Applications 27 (2015), no. 2, 179--197. doi:10.1216/JIE-2015-27-2-179.

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