Inverse Boundary Problems

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PI: Gunther Uhlmann, University of WashingtonDMS-0245414------------------------------------------------------------------------- ABSTRACTThe proposer will address several fundamental inverse problems arising in medical imaging, geophysics, and optical imaging. The systems of equations modeling the physical situation at hand are the conductivity equation, a class of Hamiltonian equations in phase space, the system of elastodynamics and Maxwell's equations. The inverse problems consist in determining the interior conductivity, the Hamiltonian system, the elastic and electromagnetic parameters, respectively, in the interior of a body or object by making measurements at the boundary. We propose four major topics of research. The first one is Electric Impedance Tomography. In this inverse method one attempts to determine the conductivity of a medium by making voltage and current measurements at the boundary. The second topic is the determination of a Riemannian metric (anisotropic sound speed) in the interior of a domain from the lengths of geodesics joiningpoints of the boundary (travel times). The third topic is seismic imaging. We plan to use the methods of microlocal analysis to generate wave-equation solutions that will help to image the heterogeneity and anisotropy of the Earth's upper crust. The last topic is ElectromagneticSystems and Maxwell's equations. In particular we propose to utilize the fact that electromagnetic fields can be polarized to determine electromagnetic parameters like chirality, which measures the asymmetry in the molecular makeup of a material.Inverse problems arise in all fields of sciences and applications where causes for a desired or observed effect are to be determined. We know about the interior structure of the Earth by solving inverse acoustic problems, the structure of DNA from solving inverse X-ray diffractionproblems, and the structure of the atom and its constituents from studying the scattering when materials are bombarded with particles. There are numerous other examples of inverse problems of practical importance. Material science provides one class of examples. One important task is tofigure out what is inside a material without destroying it in the process. Medical diagnosis provides another area where inverse problems are of vital importance. X-ray tomography and CAT scans, which are used in everyday life as diagnostic tools, are particularly familiar examples of inverse methods. In X-ray tomography high frequency X-rays are sent through the body and then the density is determined from the recorded information. The mathematics needed to recover the density was developed by Radon in the early part of the century. The goal of the proposal on inverse boundary problems is to develop mathematical reconstruction formulas for several inverse problems arising in geophysics, medical imaging and optical imaging, analog to the one developed by Radon for the case of X-ray tomography.