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

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.