A nonlinear integral equation and an iterative algorithm for an inverse source problem. (English) Zbl 1323.31006

Kress, Rainer; Rundell, William

J. Integral Equations Appl. 27, No. 2, 179-197 (2015).

Summary: 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 [Inverse Probl. 21, No. 4, 1207–1223 (2005; Zbl 1086.35139)] 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.

MSC:

31A25	Boundary value and inverse problems for harmonic functions in two dimensions
45Q05	Inverse problems for integral equations
49N45	Inverse problems in optimal control

Keywords:

inverse Dirichlet problem; inverse algorithm

References:

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