Stabilité et instabilité dans les problèmes inverses

Abstract : In this thesis we focus on stability and instability issues in some classical inverse problems for the Schrödinger equation and the acoustic equation in dimension d>=2. The problems considered are the Gel'fand inverse boundary value problem, the nearfield and the far-field inverse scattering problems. Stability and instability results presented in the thesis complement each other and contribute to a better understanding of the nature of the aforementioned problems. In particular, we prove new global stability estimates which explicitly depend on coefficient regularity and energy. In addition, we consider the inverse boundary value problem for the Schrödinger equation at fixed energy with boundary measurements represented as the impedance boundary map (or Robin-to-Robin map). We prove global stability estimates for determining potential from boundary measurements in this impedance representation. Moreover, similar techniques also give a global reconstruction procedure for this problem.
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Submitted on : Sunday, December 1, 2013 - 9:57:33 PM
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Mikhail Isaev. Stabilité et instabilité dans les problèmes inverses. Equations aux dérivées partielles [math.AP]. Ecole Polytechnique X, 2013. Français. ⟨pastel-00912298⟩

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