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

本研究将集中在偏微分方程反问题的解析和数值方面，特别是：1)半线性椭圆方程的反“源”问题，以及2)空间变系数线性椭圆方程的反“系数”问题。在这两种情况下用于重建的数据都由超定（狄利克雷或诺伊曼）边界数据组成。工作的分析部分涉及诸如可识别性和持续依赖性等问题。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将继续研究界以及各向同性和各向异性情况之间的关系。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底层PDE解的结构信息。该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）设备的磁诊断解释，2)医学阻抗成像，以及3)机械部件的无损检测。部分研究是关于确定所提出的边界数据对各种重建的充分性（证明唯一性和连续依赖性结果）。研究的另一部分涉及有效算法的设计，例如使用真实实验数据检测和定位裂缝。博士后研究人员，研究生，甚至一些高级本科生都将积极参与研究的各个方面。

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.