Imaging of nonlinear materials via the Monotonicity Principle

The topic of inverse problems, related to Maxwell's equations, in the presence of nonlinear materials is quite new in literature. The lack of contributions in this area can be ascribed to the significant challenges that such problems pose. Retrieving the spatial behaviour of some unknown physical property, starting from boundary measurements, is a nonlinear and highly ill-posed problem even in the presence of linear materials. And the complexity exponentially grows when the focus is on nonlinear material properties. Recently, the Monotonicity Principle has been extended to nonlinear materials under very general assumptions. Starting from the theoretical background given by this extension, we develop a first real-time inversion method for the inverse obstacle problem in the presence of nonlinear materials. The Monotonicity Principle is the foundation of a class of non-iterative algorithms for tomography of linear materials. It has been successfully applied to various problems, governed by different PDEs. In the linear case, MP based inversion methods ensure excellent performances and compatibility with real-time applications. We focus on problems governed by elliptical PDEs and, as an example of application, we treat the Magnetostatic Permeability Tomography problem, in which the aim is to retrieve the spatial behaviour of magnetic permeability through boundary measurements in DC operations. In this paper, we provide some preliminary results giving the foundation of our method and extended numerical examples.