Generalised Polarisation Tensors for Maxwell's Equations

通过测量电磁场来探测隐藏目标具有许多重要的应用，其中包括通过定位异物来改善食品安全，使用医学成像来发现癌性肿瘤，探测地雷和武器，以及应用地质调查来寻找新的含水层。根据应用，测量的电磁场可能与低频磁感应、中频无线电波传播、微波辐射或更高频率的可见光、紫外线和X射线有关。从测量的电磁场中确定隐藏目标的形状、位置和材料特性，需要求解“麦克斯韦反问题”。这个过程在数学和计算上都具有挑战性。在这个项目中，我们将组建一个由国际科学家、数学家和工程师组成的团队，他们将开发新的数学结果，描述电磁场在不同频率范围内与目标的不同距离上的行为。反过来，这将导致使用称为“广义偏振张量”的数学构造对目标进行单独分类。这些结果将为下一代逆解算法开辟可能性，该算法能够从电磁场测量中快速准确地检测隐藏目标。

The detection of hidden targets from measurements of electromagnetic fields has many important applications, which include improving food safety by locating foreign objects, using medical imaging to find cancerous tumours, detecting landmines and weapons as well as in the application of geological surveys to find new aquifers. Depending on the application, the measured electromagnetic fields might relate to low frequency magnetic induction, medium frequency radio wave propagation, microwave radiation or higher frequency visible light, ultraviolet and x-rays. To determine the shape, location and material properties of hidden target from the measured electromagnetic fields requires the solution of the "Maxwell inverse problem". This process is mathematically and computationally challenging.In this project, we will assemble a team of international scientists, mathematicians and engineers who will develop new mathematical results that describe how electromagnetic fields behave at different distances from the target across a range of frequencies. This, in turn, will lead to the individual classification of targets using a mathematical construction called a "generalised polarisation tensor". These results will open up possibilities for the next generation of inverse solution algorithms, which are capable of the rapid and accurate detection of hidden targets from electromagnetic field measurements.