磁感应层析成像的单调性原理
The Monotonicity Principle for Magnetic Induction Tomography
AI总结：
针对磁准静态极限下由自由响应重构电导率的反问题，本文通过模态表示感应电流密度、分析变量分离得到的椭圆特征值问题，证明了电阻率与自由响应时间常数的单调关系，为磁感应层析成像的非迭代实时成像提供基础。
摘要：
本文研究的反问题是在磁准静态（MQS）极限下，根据系统的自由响应重构电导率。MQS极限对应一个扩散偏微分方程（PDE）。在该框架下，单调性原理（Monotonicity Principle，简称MP）发挥着关键作用，即未知材料属性与可测量的自由响应之间存在单调关系。MP是非迭代实时成像方法的基础。在由不同性质的PDE支配的诸多不同物理问题中，都已发现单调性原理的存在。尽管其具有相当的普适性，但每一种不同的物理/数学语境都需要找到满足MP的合适算子，为此有必要开发针对特定框架的专用数学方法。本文证明了MQS系统中，电阻率与表征自由响应的时间常数之间存在单调关系。核心结果是通过模态表示来表征感应电流密度，主要结果基于对通过变量分离得到的椭圆特征值问题的分析。
英文摘要：
The inverse problem treated in this article consists in reconstructing the electrical conductivity from the free response of the system in the magneto-quasi-stationary (MQS) limit. The MQS limit corresponds to a diffusion PDE. In this framework, a key role is played by the Monotonicity Principle, that is a monotone relation connecting the unknown material property to the (measured) free-response. MP is relevant as basis of noniterative and real-time imaging methods. Monotonicity Principles have been found in many different physical problems governed by PDEs of different nature. Despite its rather general nature, each different physical/mathematical context requires to discover the proper operator showing MP. For doing this, it is necessary to develop ad-hoc mathematical approaches tailored on the specific framework. In this article, we prove a monotonic relationship between the electrical resistivity and the time constants characterizing the free-response for MQS systems. The key result is the representation of the induced current density through a modal representation. The main result is based on the analysis of an elliptic eigenvalue problem, obtained from separation of variables.
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