Inverse Problems for Partial Differential Equations

拟议的工作有四个主要目标。(1)第一个目标是进一步推进映射隐身的相关工作。将特别强调所提议的超材料（用于隐身）的物理相关性，以及改进近似方案的相对隐身率的潜力。在物理相关性的背景下，目标是研究满足所谓的Kramers-Kronig关系的元材料（从因果关系考虑中自然产生）。最初，这些研究将集中在材料上，这是由所谓的德鲁德-洛伦兹模型控制的。此外，还计划对完全匹配层（PML）技术和映射隐身技术之间的相似性进行仔细研究。(2)第二个目标是从对多个边界载荷的响应中更好地估计非均匀性的存在，特别是它们的体积分数。目的是扩展早期的低体积分数结果，获得少量的边界测量（n个维度的n个测量）。(3)第三个目标是研究一个反问题的唯一性和稳定性，在这个反问题中，人们试图从单个（诺伊曼）边界测量中识别（非线性）电流通量。相关的边值问题是由磁流体动力学平衡方程驱动的。(4)第四个也是最后一个目标是研究与腐蚀和氧化建模相关的静态和动态问题。这些研究将从显式解的构造，到边界爆破的研究，以及有限时间爆破与所有时间存在的标准。这很有可能产生更广泛的影响。在“测绘隐形”领域，考虑到（物理上相关的）用于制造隐形的超材料的限制，人们期望获得的这种结果对于确定在实践中可以实现的“近隐身”程度将非常有用。人们可以很好地想象预期结果的类型与近似隐形斗篷的实际物理结构之间的相互作用。对小不均匀性的估计/检测工作已经导致了用于缺陷检测和医学成像的算法的发展。对于体积分数，改进后的边界将有类似的潜在应用。虽然在磁流体动力学背景下对电通量可识别性的研究将主要是理论上的，但很明显，稳定性特性的研究以及使用额外的线积分测量的好处，在托卡马克装置中的等离子体控制方面可能具有非常实际的应用。

The proposed work has four major goals. (1) The first goal is to further advance the work concerning cloaking-by-mapping. Particular emphasis will be placed on the physical relevance of the proposed meta-materials (used for the cloak), and on the potential for improvement in the relative cloaking rates of the approximate schemes. In the context of physical relevance, the goal is to study meta- materials that satify the socalled Kramers-Kronig relations (which naturally emerge from causality considerations). Initially these studies will focus on materials, which are governed by the socalled Drude-Lorentz model. It is also planned to perform a careful study of the similarities between the technique of perfectly matched layers (PML) and cloaking-by-mapping. (2) The second goal is to better estimate the presence of inhomogeneities, in particular their volume fraction, from the response to multiple boundary loads. The aim is to extend earlier low volume fraction results, obtained for small number of boundary measurements (n measurements in n dimensions). (3) The third goal is to study the uniqueness and stability of an inverse problem, in which one seeks to identify a (nonlinear) current flux from a single (Neumann) boundary measurement. The associated boundary value problem is motivated by the equations of magnetohydrodynamic equilibria. (4) The fourth and final goal is to study static and dynamic problems associated with corrosion and oxidation modelling. These studies will range from the construction of explicit solutions, to the study of boundary blow-up, and criteria for finite time blow-up vs. existence for all time.There are significant possibilities for broader impact. In the area of "cloaking-by-mapping", the kind of results, one can expect to obtain, will be extremely useful in determining the degree of "near invisibility" that is in practice achievable, given (physically relevant) limitations on the meta-materials, used to construct the cloaks. One can well imagine an interplay between the type of results expected, and the actual physical construction of approximate invisibility cloaks. The work on the estimation/detection of small inhomogeneities has already led to the development of algorithms of use in flaw detection and medical imaging. The improved bounds, for volume fractions, will have similar potential applications. While the studies of the identifiability of the electric flux in the context of magnetohydrodynamics will be mostly theoretical, it is clear that the study of stability properties, and of the benefits of using additional line-integral measurements, could have very practical applications in terms of the control of plasmas in Tokamak devices.