  In this paper we present a first non-iterative imaging method for nonlinear
materials, based on Monotonicity Principle. Specifically, we deal with the
inverse obstacle problem, where the aim is to retrieve a nonlinear anomaly
embedded in linear known background.
  The Monotonicity Principle (MP) is a general property for various class of
PDEs, that has recently generalized to nonlinear elliptic PDEs. Basically, it
states a monotone relation between the point-wise value of the unknown material
property and the boundary measurements. It is at the foundation of a class of
non-iterative imaging methods, characterized by a very low execution time that
makes them ideal candidates for real-time applications.
  In this work, we develop an inversion method that overcomes some of the
peculiar difficulties in practical application of MP to imaging of nonlinear
materials, preserving the feasibility for real-time applications. For the sake
of clarity, we focus on a specific application, i.e. the Magnetostatic
Permeability Tomography where the goal is retrieving the unknown (nonlinear)
permeability by boundary measurements in DC operations. This choice is
motivated by applications in the inspection of boxes and containers for
security.
  Reconstructions from simulated data prove the effectiveness of the presented
method.
