Mathematical Problems in Low Frequency Electromagnetic Inversion and in Inverse Scattering in Random Media

我们考虑了两个反问题的理论和数值研究：(A)低频电磁反问题，在有界区域内，我们在给定Neumann到Dirichlet映射的情况下，在有界域内寻找椭圆型方程组的未知系数(电导率/介电常数)。我们探讨了变分原理在解决这类问题中的应用。特别是，我们希望开发新的变分重建算法，并研究分辨率极限(可区分性)。在数值反演中，进行适当的离散化是至关重要的。我们提出了问题的有限差分离散，在最优网格上。我们已经证明，在一维(Sturm-Liouville)问题中，最优网格给出了稳定和有效的反演算法。我们希望进一步研究最优网格，并将其推广到更高的维度。(B)随机介质中的逆散射，其中我们希望成像掩埋在杂波中的目标的反射率，我们将其建模为随机介质。我们对遥感制度很感兴趣，因为不均匀包裹体对波的多路径(多次散射)有很大的影响。这项拟议的工作从仔细研究波在随机介质中的传播开始，它致力于开发统计稳定的成像算法，它可以提供可靠的图像，而不依赖于人们对杂波细节的缺乏了解。我们考虑了两个逆问题的理论和数值研究：(A)第一个问题考虑了通过测量物体表面的电流和电压或电场和磁场，恢复物体的电导率和介电常数等性质。因为不同的材料表现出不同的电学性质，所以每当我们想要通过收集物体外围的数据来推断物体的内部结构时，这些问题就变得重要起来。应用的例子有：(1)医学：检测肺血栓，监测心脏功能和血流，检测乳腺肿瘤等。(2)环境科学：检测地下储罐泄漏，监测地下流动等。(3)材料无损检测：检测金属中的腐蚀、裂缝和空洞等。我们的研究集中在使用最先进的变分技术和优化网格(参数化)，以获得高效和可靠的人体内未知电特性的恢复。(B)第二个问题考虑杂波中目标的探测和成像，通过有源天线阵列(换能器)在介质中发送探测信号并记录散射回波。到目前为止，对杂波成像还不是很清楚，但它在超声波成像、陆地或浅水地雷探测、地面或树叶穿透雷达等方面有重要的应用。我们的杂波成像方法是基于随机介质中波传播的知识，它考虑了统计稳定成像算法的发展，这些算法给出可靠的结果，与杂波的不确定性无关。

We consider theoretical and numerical studies of two inverse problems:(a) Low frequency electromagnetic inversion, where we seek unknowncoefficients (electrical conductivity/permittivity) in ellipticsystems of equations, inside a bounded domain, given the Neumann toDirichlet map at the boundary. We explore the use of variationalprinciples in the solution of such problems. In particular, we wishto develop new, variational reconstruction algorithms and to studyresolution limits (distinguishability). In numerical inversion, havinga proper discretization is paramount. We propose a finite differencediscretization of the problem, on optimal grids. We have demonstratedthat optimal grids give stable and efficient inversion algorithms, inone dimensional (Sturm-Liouville) problems. We wish to study furtherthe optimal grids and to extend them to higher dimensions. (b) Inverse scattering in random media, where we wish to image thereflectivity of targets buried in clutter, which we model as a randommedium. We are interested in remote sensing regimes, with significantmultipathing (multiscattering) of the waves, by the inhomogeneities inclutter. The proposed work starts with a close look at wave propagation in random media and it strives to develop statisticallystable imaging algorithms, which give reliable images, independent ofone's lack of knowledge of the details of the clutter.We consider theoretical and numerical studies of two inverse problems:(a) The first problem considers the recovery of properties such as theelectrical conductivity and permittivity of a body, given measurementsof electric currents and voltages, or the electric and magnetic fields, at the surface of the body. Because different materials display different electrical properties, these problems are important whenever we wish to infer the internal structure of a body, by gathering data at its periphery. Examples of applications are: (1) In medicine: for detection of pulmonary emboli, monitoring of heart function and blood flow, detection of breast tumors, etc. (2) In environmental sciences: for detection of leaks in underground tanks, monitoring of underground flows, etc. (3) Nondestructive testing of materials: detection of corrosion, cracks and voids in metals,etc. Our research focuses on using state of the art variational techniques and optimal grids (parametrizations), to obtain efficient and reliable recoveries of the unknown electrical properties inside the body. (b) The second problem considers the detection and imaging of targetsin clutter, via active arrays of antennas (transducers) which sendprobing signals in the medium and record the scattered echoes. Imagingin clutter is not well understood, so far, but it has importantapplications in ultrasound imaging, land or shallow water minedetection, ground or foliage penetrating radar, etc. Our approach toimaging in clutter is based on knowledge of wave propagation in randommedia and it considers the development of statistically stable imagingalgorithms, which give reliable results, independent of one'suncertainty of the clutter.