Mathematical Problems for Nonlinear Inversion in Intermediate and High Contrast Media

[9971209]这个研究项目将考虑椭圆型偏微分方程的逆问题，这些方程的系数在解的域上有很大的变化。具体来说，重点将放在成像电导率和介电常数的介质，给定低频电磁场测量。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。研究目标是建立一套连贯的算法和分析工具，用于成像中等和高对比度的介质，超越通常的线性化程序，这显然是不充分的。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造影剂。2)基于我们在高对比度介质中对时谐电磁场的新渐近研究，开发成像算法。(3)将渐近理论推广到含有高体积分数刚性夹杂的介质的线弹性问题。异质介质的电学特性成像在地下污染物检测等环境研究中有着重要的应用。由于已知地下电导率和介电常数变化很大，可达数量级，因此研究的重点是中、高对比度介质。对地下流体羽流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或电磁测量。对收集到的数据进行处理，以便估计地下良导体的位置及其大小。对于电直流电的测量，已经做了大量的工作。相比之下，对电磁反问题的研究较少。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，并计划将这些研究应用于反演。结果是有希望的，并将带来显着的改进，以现有的成像算法的现阶段。

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.