Uniqueness, reconstruction, stability for some two-dimensional inverse problems

Abstract : In this thesis some inverse boundary value problems in two dimensions are studied. The problems under consideration are the Calder'on problem and the Gel'fand-Calderon problem in the single and multi-channel case, i.e. matrix-valued case: the latter can be seen, in particular, as a non-overdetermined approximation of the three dimensional case. We begin with some results for the anisotropic Calderon problem: a new formulation of the uniqueness result on the plane is presented, as well as the first global uniqueness on two-dimensional surfaces with boundary. Next, we prove some new global stability estimates for the Gel'fand-Calderon problem in the single and multi-channel case. Similar techniques give also a global reconstruction procedure for the same problem in the multi-channel case. A rapidly converging approximation algorithm for the multi-channel Gel'fand-Calderon problem is presented afterwards: this is mostly inspired by results of the multi-dimensional inverse scattering theory. Finally we present new global stability estimates for the two aforementioned problems which explicitly depend on regularity and energy.
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Submitted on : Monday, December 3, 2012 - 11:54:38 AM
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Matteo Santacesaria. Uniqueness, reconstruction, stability for some two-dimensional inverse problems. Analysis of PDEs [math.AP]. Ecole Polytechnique X, 2012. English. ⟨pastel-00759992⟩

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