Mathematical Problems for Nonlinear Inversion in Intermediate and High Contrast Media

9971209Borcea这个研究项目将考虑椭圆偏微分方程的反问题，其系数在解的域中变化很大。具体来说，重点将是成像的电导率和介电常数的介质，给定的低频电磁场测量。这样的逆问题是高度非线性的，并提出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，成像中间和高对比度的媒体，超越通常的线性化程序，显然是不够的。为了实现这一目标，已经确定了以下研究方向：1）在我们的算法中使用最先进的最小变分原理来成像中间对比剂。2)基于我们对高对比度介质中时谐电磁场的新渐近研究，开发成像算法。3)将渐近理论扩展到具有高体积分数刚性夹杂物的介质的线性弹性问题。成像非均匀介质的电特性在环境研究中有重要应用，如地下污染物检测。研究集中在中间和高对比度的媒体，因为已知的事实，地下的电导率和介电常数可以有很大的变化，高达数量级。对地下流体羽流的形状进行成像的一种实用方法是在地球表面和钻孔中进行电气或电磁测量。对收集的数据进行处理，以估计地下良导体的位置及其大小。对于直流电测量的情况，已经做了大量的工作。相对而言，电磁场逆问题的研究较少。大多数可用的结果适用于低对比度介质。最近，我们已经做了高对比度介质中的电磁场的研究，并计划将这些研究用于反演。结果是有希望的，并应带来显着的改善，目前阶段的可用成像算法。

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.