Inverse problems, which are related to Maxwell's equations, in the presence of nonlinear materials is a quite new topic in the 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, from boundary measurements, is a nonlinear and highly ill-posed problem even in the presence of linear materials. Furthermore, this complexity grows exponentially in the presence of nonlinear materials. In the tomography of linear materials, the Monotonicity Principle (MP) is the foundation of a class of non-iterative algorithms able to guarantee excellent performances and compatibility with real-time applications. Recently, the MP has been extended to nonlinear materials under very general assumptions. Starting from the theoretical background for this extension, we develop a first real-time inversion method for the inverse obstacle problem in the presence of nonlinear materials. The proposed method is intendend for all problems governed by the quasilinear Laplace equation, i.e. static problems involving nonlinear materials. In this paper, we provide some preliminary results which give the foundation of our method and some extended numerical examples.

翻译：与麦克斯韦方程组相关的逆问题在非线性材料存在时是文献中一个相当新颖的课题。该领域研究成果匮乏可归因于此类问题所带来的巨大挑战。即使在线性材料存在的情况下，通过边界测量反演未知物理性质的空间分布也是一个非线性的、高度不适定问题。此外，当存在非线性材料时，这种复杂性会呈指数级增长。在线性材料层析成像中，单调性原理是一类非迭代算法的基础，这类算法能够保证优异的性能并与实时应用兼容。最近，在非常一般的假设下，单调性原理已被推广到非线性材料。基于这一推广的理论背景，我们开发了首个针对非线性材料存在时逆障碍问题的实时反演方法。所提出的方法旨在适用于所有由拟线性拉普拉斯方程支配的问题，即涉及非线性材料的静态问题。本文提供了一些初步结果，这些结果构成了我们方法的基础，并给出了扩展的数值实例。