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，其中我们在有界域中的椭圆方程系统中寻求未知系数（电导率/介电常数），给定边界处的诺伊曼到狄利克雷映射。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限制（可分辨性）。在数值反演中，适当的离散化是至关重要的。我们在最优网格上提出了问题的有限差分离散化方法。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，并将其扩展到更高的维度。(b)随机介质中的逆散射，其中我们希望成像埋在杂波中的目标的反射率，我们将其建模为随机介质。我们感兴趣的遥感制度，与显著多路径（多重散射）的波，由不均匀夹杂。提出的工作从仔细观察随机介质中的波传播开始，并努力开发统计稳定的成像算法，这些算法可以提供可靠的图像，而不需要缺乏对杂波细节的了解。我们考虑两个反问题的理论和数值研究：(a)第一个问题考虑在给定物体表面的电流和电压或电场和磁场的测量下，物体的电导率和介电常数等性质的恢复。由于不同的材料表现出不同的电学性质，每当我们希望通过收集物体外围的数据来推断物体的内部结构时，这些问题都很重要。应用实例有：(1)医学领域：用于肺栓塞的检测、心功能和血流的监测、乳腺肿瘤的检测等；(2)环境科学领域：用于地下储罐泄漏的检测、地下流量的监测等；(3)材料的无损检测：金属的腐蚀、裂纹和空隙的检测等。我们的研究重点是使用最先进的变分技术和最优网格（参数化），以获得体内未知电特性的有效和可靠的恢复。(b)第二个问题考虑在杂波中探测和成像目标，通过在介质中发送探测信号并记录散射回波的有源天线（换能器）阵列。杂波成像技术目前还没有得到很好的理解，但它在超声成像、陆地或浅水水雷探测、地面或树叶穿透雷达等方面有着重要的应用。我们在杂波中成像的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，它给出了可靠的结果，独立于杂波的不确定性。

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.