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

我们考虑两个反问题的理论和数值研究：（a）低频电磁反演，在有界区域内，给定边界处的Neumann-Dirichlet映射，在椭圆方程组中寻找未知系数（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，变分重建算法和研究分辨率限制（可扩展性）。在数值反演中，适当的离散化是至关重要的。 我们提出了一个有限差分离散化的问题，最佳网格。 我们已经证明，最佳网格提供稳定和有效的反演算法，一维（Sturm-Liouville）问题。 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)在随机介质中的逆散射，我们希望图像的反射率的目标埋在杂波，我们作为一个随机介质模型。我们感兴趣的遥感制度，与显着multipathing（多散射）的波，由inhomogeneities incluter。 建议的工作开始仔细看看波在随机介质中的传播，它努力开发的criticalstable成像算法，这给可靠的图像，独立于一个人的缺乏知识的细节的杂波。我们考虑两个逆问题的理论和数值研究：（a）第一个问题考虑恢复的属性，如身体的电导率和介电常数，给定measurementsof电流和电压，或电场和磁场，在身体的表面。由于不同的材料显示出不同的电学特性，因此，每当我们希望通过收集物体外围的数据来推断其内部结构时，这些问题都很重要。 应用实例如下：（1）在医学方面：用于检测肺栓塞、监测心脏功能和血流、检测乳腺肿瘤等。（2）环境科学：用于检测地下储罐泄漏、监测地下流量等。（3）材料的无损检测：我们的研究重点是使用最先进的变分技术和最佳网格（参数化），以获得体内未知电特性的有效和可靠的恢复。 (b)第二个问题考虑了目标杂波的检测和成像，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，杂波中的成像还没有得到很好的理解，但它在超声成像、陆地或浅水地雷探测、地面或叶面穿透雷达等方面有重要的应用。我们的杂波中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，这些算法给出了可靠的结果，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.