Analytical and computational studies of direct and inverse boundary value problems for PDEs

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Abstract, Award DMS-0307119Michael Vogelius, Rutgers UniversityTitle: Analytical and computational studies of direct and inverse boundary value problems for PDEsProject Abstract:A main object of this project is to delvelop very detailed information about the response of a system (a PDE boundary value problem) to the presence of internal inhomogeneities (defects), the spacing of these inhomogeneities, or changes in the boundary conditions. One important set of tools to obtainthis information are rigorously verfied asymptotic expansions. In the case of changing, nonlinear boundary conditions another crucial component is the study of mechanisms that govern the development of (near-)singularities. The second main theme is then the application of this information to effectively obtain knowledge about (reconstruct) the parameters of the system (e.g., the location, and the character of the inhomogeneities) from data measured on an accessible part of boundary, or in the ``farfield''. Initially the focus will be on steady state and time harmonic situations, with consideration of scalar equations as well as systems (in particular Maxwell's Equations). Later on fully time-dependent cases will also be studied. The work has an essential computational component. The particular applications that will be investigated concern 1) anti-personnel mine detection, using Ground Penetrating Radar(GPR) data, and 2) corrosion/oxidation imaging (and control) using partial voltage and current data. But many other applications of significant interest come to mind, for instance in the field of medical tomography. It is expected that the combination of analysis, asymptotic analysis and computational work contained in this project will allow for the design of detection (and location) algorithms that are more precise (less sensitive to noise), and much faster than those of a general purpose nature. For reconstructions it is envisaged to work on experimental as well as synthetic data.I intend to organize two small workshops in the second and third year (of the period covered by this grant). The first will be devoted to mathematicaland computational aspects of corrosion/oxidation modeling. The second will be devoted to GPR mine-detection. The participants will be mathematicians and engineers.