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.

翻译：反问题中涉及麦克斯韦方程组及非线性材料的研究在文献中尚属新兴领域。该领域文献匮乏的原因在于此类问题带来的巨大挑战。即便在线性材料情况下，通过边界测量反演未知物理属性的空间分布已属于非线性和高度病态问题；当关注非线性材料特性时，其复杂度更呈指数级增长。近期，单调性原理在非常通用的假设条件下被扩展至非线性材料。基于该扩展的理论基础，我们首次针对非线性材料中的逆障碍物问题，开发出一种实时反演方法。单调性原理是线性材料层析成像非迭代算法的基础，已成功应用于不同偏微分方程主导的多种问题。在线性情形下，基于单调性原理的反演方法展现出优异性能与实时应用的兼容性。本文聚焦于椭圆型偏微分方程主导的问题，并以静磁磁导率层析成像问题作为应用实例——该问题的目标是通过直流操作中的边界测量来反演磁导率的空间分布。我们提供了初步结果作为该方法的基础，并给出了扩展数值算例。