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

这项研究将集中在偏微分方程反问题的分析和数值方面，特别是：1)半线性椭圆型方程的反“源”问题，以及2)具有空间变系数的线性椭圆型方程的反“系数”问题。在这两种情况下用于重建的数据都由超定(Dirichlet或Neumann)边界数据组成。这项工作的分析部分涉及诸如可辨识性和连续依赖性等问题。对于逆“源问题”，PI将进一步研究与著名的Schiffer(或Pompeiu)猜想的关系；对于逆“系数”问题，PI将继续他的边界以及各向同性和非自转情况之间的关系的研究。数值工作将致力于设计有效的重建方法，这些方法最大程度地依赖于关于基本偏微分方程组的解的结构信息。这项研究直接应用于几个重要的实际问题，例如：1)托卡马克(聚变)装置的磁学诊断解释，2)医学阻抗成像，3)机械部件的无损检测。研究的一部分涉及确定所提出的边界数据对于各种重建的充分性(证明唯一性和连续依赖结果)。研究的另一部分涉及设计有效的算法，例如使用真实的实验数据来检测和定位裂缝。博士后研究人员、研究生，甚至一些高级本科生将积极参与这项研究的各个方面。

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.