Mathematical Sciences: Analytical & Numerical Aspects of Inverse Problems for Differential Equations

这项研究将集中在分析和数值方面 偏微分方程的反问题，并在 特别是：1）半线性椭圆型方程的反“源”问题 方程，和2）反“系数”问题的线性 变系数椭圆方程 数据 在这两种情况下，用于重建的 （Dirichlet或Neumann）边界数据。 分析组分 的工作涉及这样的问题，如可识别性和 持续依赖。 对于逆“源问题”，PI 此外，将调查与众所周知的 希弗（或庞培）猜想;为反“系数” 问题的PI将继续他的调查范围，以及 作为各向同性和各向异性情况之间的关系。的 数值计算工作将致力于设计有效的 重建方法，在最大程度上可能依赖于 关于基础问题的解决方案的结构信息 PDE。 这项研究直接应用于几个重要的 实际问题，例如：1）解释 托卡马克（聚变）装置的磁诊断，2）医疗 阻抗成像，和3）机械的无损检测 零件. 部分研究涉及确定 各种拟议边界数据的充分性 重建（证明唯一性和连续依赖性 结果）。 研究的另一部分涉及设计 有效的算法，例如用于检测和定位 用真实的实验数据进行裂纹的计算。 会有一个活跃的 博士后研究人员以及研究生的参与 学生，甚至希望是一些高年级的本科生 在研究的各个方面。

This research will focus on analytical and numerical aspects of inverse problems for partial differential equations, and in particular: 1) inverse "source" problems for semilinear elliptic equations, and 2) inverse "coefficient" problems for linear elliptic equations with spatially varying coefficients. The data used for reconstruction in both cases consists of overdetermined (Dirichlet or Neumann) boundary data. The analytical component of the work concerns such questions as identifiability and continuous dependence. For the inverse "source problem", the PI furthermore will investigate the relation to the well known Schiffer (or Pompeiu) conjecture; for the inverse "coefficient" problem the PI will continue his investigation of bounds as well as the relation between the isotropic and aniostropic cases. The numerical work will be devoted to the design of effective reconstruction methods which to the largest extent possible rely on structural information about the solutions of the underlying PDE. The research has direct applications to several important practical problems, for instance: 1) the interpretation of magnetic diagnostics for Tokamak (fusion) devices, 2) medical impedance imaging, and 3) nondestructive testing of mechanical parts. Part of the research is concerned with determining the sufficiency of the proposed boundary data for the various reconstructions (proving uniqueness and continuous dependence results). Another part of the research involves the design of effective algorithms, for instance for the detection and location of cracks using real experimental data. There will be an active involvement of post-doctoral researchers as well graduate students and hopefully even some advanced undergraduate students in various aspects of the research.