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

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.


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


Zbl 1086.35139
Full Text: DOI Euclid


[1] F. Cakoni, Y. Hu and R. Kress, Simultaneous reconstruction of shape and generalized impedance functions in electrostatic imaging , Inv. Prob. 30 (2014), 105009. · Zbl 1305.65222 · doi:10.1088/0266-5611/30/10/105009
[2] H.W. Engl, M. Hanke and A. Neubauer, Regularization of inverse problems , Kluwer Academic Publisher, Dordrecht, 1996. · Zbl 0859.65054
[3] M. Hanke and W. Rundell, On rational approximation methods for inverse source problems , Inv. Prob. Imag. 5 (2011), 185-202. · Zbl 1215.35166 · doi:10.3934/ipi.2011.5.185
[4] F. Hettlich and W. Rundell, Iterative methods for the reconstruction of an inverse potential problem , Inv. Prob. 12 (1996), 251-266. · Zbl 0858.35134 · doi:10.1088/0266-5611/12/3/006
[5] V. Isakov, Inverse problems for partial differential equations , 2nd. edition, Springer, New York, 2005. · Zbl 0908.35134
[6] O. Ivanyshyn and R. Kress, Nonlinear integral equations for solving inverse boundary value problems for inclusions and cracks , J. Integral Equations Appl. 18 (2006), 13-38; Corrigendum: J. Integral Equations Appl. 22 (2010), 647-649. · Zbl 1139.45003 · doi:10.1216/jiea/1181075363
[7] T. Johansson and B. Sleeman, Reconstruction of an acoustically sound-soft obstacle from one incident field and the far-field pattern , IMA J. Appl. Math. 72 (2007), 96-112. · Zbl 1121.76059 · doi:10.1093/imamat/hxl026
[8] B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative regularization methods for nonlinear ill-posed problems , de Gruyter, Berlin, 2008. · Zbl 1145.65037 · doi:10.1515/9783110208276
[9] A. Kirsch, An introduction to the mathematical theory of inverse problems , 2nd. edition, Springer, New York, 2011. · Zbl 1213.35004 · doi:10.1007/978-1-4419-8474-6
[10] R. Kress, Integral equations , 3rd. edition, Springer, New York, 2014. · Zbl 1328.45001 · doi:10.1007/978-1-4614-9593-2
[11] —-, A collocation method for a hypersingular boundary integral equation via trigonometric differentiation , J. Integral Equations Appl. 26 (2014), 197-213. · Zbl 1310.65169 · doi:10.1216/JIE-2014-26-2-197
[12] R. Kress and W. Rundell, Nonlinear integral equations and the iterative solution for an inverse boundary value problem , Inv. Prob. 21 (2005), 1207-1223. · Zbl 1086.35139 · doi:10.1088/0266-5611/21/4/002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.