Mathematical Problems for Nonlinear Inversion in Intermediate and High Contrast Media

这个研究项目将考虑系数在解的区域内变化很大的椭圆型偏微分方程逆问题。具体地说，重点将放在给出低频电磁场测量的情况下，成像介质的电导率和介电常数。这种反问题是高度非线性的，并提出了困难的分析和计算问题。研究的目标是建立一套连贯的算法和分析工具，用于中高对比度介质的成像，超越通常的线性化程序，这显然是不够的。为了实现这一目标，我们确定了以下研究方向：1)在成像中间造影剂的算法中使用最先进的最小变分原理。2)基于我们对高对比度介质中时间谐波电磁场的新的渐近研究，发展了成像算法。3)将渐近理论推广到刚性夹杂体积分数较高的介质中的线弹性问题。非均匀介质的成像电学特性在地下污染物检测等环境研究中具有重要的应用。由于地下电导率和介电常数可能有很大的变化，可达数量级，因此研究重点放在中高对比度介质上。成像地下流体羽流形状的一种实用方法是在地球表面和钻孔中进行电磁测量。对收集的数据进行处理，以估计地下良导体的位置和大小。对于华盛顿特区的电气案件，已经做了大量的工作。测量。相对而言，对电磁场逆问题的研究相对较少。大多数可用的结果适用于低对比度介质。最近，我们对高对比度介质中的电磁场进行了研究，并计划将这些研究用于反演。结果是有希望的，应该会给当前阶段的现有成像算法带来显着的改进。

9971209BorceaThis research project will consider inverse problems for elliptic partial differential equations with coefficients that vary substantially in the domain of the solution. Specifically, the focus will be on imaging electrical conductivity and permittivity of a medium, given low frequency electro-magnetic field measurements. Such inverse problems are highly nonlinear and pose difficult analytical and computational questions. The research goal is to build a coherent set of algorithms and analytic tools for imaging intermediate and high contrast media, that go beyond the usual linearization procedure that is clearly inadequate. Towards this goal, the following directions of research have been identified: 1) Use state of the art, minimal variational principles in our algorithms for imaging intermediate contrast media. 2) Develop imaging algorithms based on our new asymptotic studies for time-harmonic electro-magnetic fields in high contrast media. 3) Extend the asymptotic theory to problems in linear elasticity, for media with a high volume fraction of rigid inclusions.Imaging electrical properties of heterogeneous media finds important applications in environmental studies such as underground contaminant detection. The research focuses on intermediate and high contrast media because of the known fact that the underground electrical conductivity and permittivity can have large variations, up to orders of magnitude. One practical way of imaging the shape of fluid plumes underground is to make electrical or electro-magnetic measurements on the surface of the Earth and in boreholes. The data gathered are processed in order to estimate both the location of underground good conductors and their magnitude. A substantial amount of work has been done for the case of electrical, d.c. measurements. Comparatively, little work has been done on the electro-magnetic inverse problem. Most of the results available apply to low contrast media. Recently, we have done studies of electro-magnetic fields in high contrast media and plan to use these studies in inversion. The results are promising and should bring significant improvement to the current stage of available imaging algorithms.