Journal of Elliptic and Parabolic Equations

, Volume 4, Issue 1, pp 177–205 | Cite as

One-iteration reconstruction algorithm for geometric inverse source problem



In this paper, we address the reconstruction of characteristic source functions \(\delta g^*\) in the elliptic partial differential equations \(-\Delta u+u=\delta g^*\) from the knowledge of the boundary measurements. We will detect the shape and location of the unknown source term from additional boundary conditions. We propose a new reconstruction method based on the Kohn–Vogelius formulation and the topological gradient method. The inverse problem is formulated as a topological optimization one. An asymptotic expansion for an energy function is derived with respect to a small topological perturbation of the source term. The unknown source is reconstructed using a level-set curve of the topological gradient. A non-iterative reconstruction procedure based on the topological sensitivity is implemented. The efficiency and accuracy of the reconstruction algorithm are illustrated by some numerical results.


Geometric inverse source problem Topological sensitivity analysis Topological optimization Kohn–Vogelius formulation 

Mathematics Subject Classification

35A15 35B20 49K40 65R32 


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© Orthogonal Publishing and Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of SciencesMonastir UniversityMonastirTunisia

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