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Title: EIT via local measurements
Significant progress has been made in the last twenty years or so in both the theoretical and applied aspects of EIT. In particular the construction of complex geometrical optics solutions has led to many results. These results have for the most part assumed that we can make the measurements on the whole boundary.
In this talk we will survey recent progress on the determination of a conductivity by making electrical measurements on part of the boundary. The speaker and A. Bukhgeim have shown that one can prove unique identifiability of the conductivity by making the measurements on particular open subsets of the boundary depending on the geometry of the domain. The method of proof uses Carleman estimates to construct appropriate complex geometrical optics solutions. We will also discuss recent joint work with H. Ammari. We prove that we can determine the conductivity in the interior by measuring the Dirichlet-to-Neumann map on arbitrary open subsets of the boundary by applying voltage potentials in the same open subset, if one knows a-priori the conductivity in a neighborhood of the boundary.