Real-time imaging methods for tomography of linear and nonlinear materials

The thesis proposes some innovative approaches for real-time tomography. Specifically, the inverse obstacle problem is considered, where the aim is to retrieve shape, position and dimension of one or more anomalies embedded in a known background by probing the system with a certain kind of electromagnetic energy and making measurements on the boundary, i.e. in a non destructive manner. First, the "classical" problem of a linear anomaly in a linear background is considered and a new real-time inversion method is proposed, which has is peculiar feature in the simple but very stable numerical implementation. Then, the complete new problem of a nonlinear anomaly embedded in a linear background is deeply investigated. Along the mathematic treatment of this problem, also in this case, a practical real-time inversion method is established. All the methods proposed are valitated by extensive numerical examples which prove the effectiveness of the proposed techniques.