A Step Into the World of Nonlinear Inverse Problems

The aim of the course is to present nonlinear inverse problems for partial differential equations with significant applications in imaging (medical and geophysical). The main prototype to be discussed, which exhibits the ill-posedness features common to many inverse problems, is the electrical impedance tomography model, from which important analytical and numerical results have emerged over the last 40 years. The course will address the ill-posed nature of inverse problems, issues of uniqueness, and estimates of continuous dependence on the data starting from an infinite number of measurements, as well as possible regularization techniques. It will then move on to recent results based on a finite number of measurements and, in this context, introduce mathematical problems still open.