Analytical and Computational Studies of Direct and Inverse Boundary Value Problems for PDEs

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The project has two major goals. The first goal is to uncover specific information about the response of various physical systems (described by boundary value problems for partial differential equations) to the presence of inhomogeneities (defects), the spacing of these inhomogeneities, or parametric changes in the boundary conditions. Emphasis is placed on developing very precise asymptotic formulas and a priori estimates that may help to assess the accuracy of these formulas. The work largely (but not exclusively) focuses on steady state and time harmonic situations, considering scalar equations as well as systems (in particular Maxwell's equations). Significant effort will be invested in accurate characterization of the response for a wide range of frequencies. The second goal is to demonstrate how to use such structural knowledge about the nature of the response, in combination with measured data (say, from an accessible part of boundary, or from the "far field") to effectively infer information about the parameters of the system (e.g., the location and the character of the inhomogeneities). Special attention will be given to methods that effectively use a wide range of frequencies to improve reconstructions.brbrThis project investigates fundamental mathematical problems that underlie nondestructive inspection of hidden aspects of objects. Three particular imaging applications concern ground penetrating radar (e.g., for anti-personnel mine detection) eddy current imaging (e.g.. for corrosion assessment), and optimal imaging (for biomedical applications). This work will form a basis for the design of detection (and location) algorithms that are more precise (less sensitive to noise) and much faster than those currently in use. The project will develop software of use to practitioners that will be made publicly available. The work may lead to the development of a theoretical framework that will be useful in many other contexts as well.