Mathematische Theorie direkter und inverser transienter Wirbelstromprobleme

瞬态（如脉冲）励磁电流产生电磁场，电磁场又在近端导体中感应电流。在数学上，这可以用偏微分方程来描述，即涡流方程，它是通过忽略麦克斯韦方程中的介电位移电流而得到的。涡流方程为抛物线-椭圆型：在绝缘区，场立即适应激励（准平稳椭圆行为），而在导电区，由于感应涡流（抛物线行为），这种适应需要一些时间。涡流效应用于远程探测导电物体（例如在地雷探测的背景下）和非侵入性地识别导体内部的缺陷（所谓的涡流检测）。用数学术语来说，这导致了根据（部分）解的知识重建涡流方程中电导率系数的反问题。在本项目中，我们旨在利用抛物-椭圆方程的统一变分理论，从理论上研究逆问题中的可辨识性问题，并推导出严格合理的重构策略。

Transient (e.g. pulsed) excitation currents generate electromagnetic fields which in turn induce electric currents in proximal conductors. Mathematically, this can be described by partial differential equations, the eddy-current equations, which are obtained by neglecting the dielectric displacement currents in Maxwell’s equations. The eddy-current equations are of parabolic-elliptic type: In insulating regions, the field instantaneously adapts to the excitation (quasistationary elliptic behaviour), while in conducting regions, this adaptation takes some time due to the induced eddy currents (parabolic behaviour). Eddy current effects are used for remotely detecting conducting objects (e.g. in the context of land mine detection) and to non-invasively identify flaws inside a conductor (so-called eddy-current testing). In mathematical terms this leads to the inverse problem of reconstructing the conductivity coefficient in the eddy current equations from (partial) knowledge of the solution(s). In the proposed project we aim to utilize a unified variational theory for the parabolic-elliptic equations to theoretically study identifiability questions in the inverse problem and derive rigorously justified reconstruction strategies.