Some inverse problems: increasing stability and drift-diffusion models

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The goal of this project is to improve the resolution of numerical algorithms for the recovery of sources, obstacles, and medium characteristics from exterior measurements of various physical fields found in biomedicine, economics, geophysics, and material science. In particular, the results will dramatically enhance the quality of a cheap, fast, and safe diagnostic method called electrical impedance tomography. Examples of specific applications include deriving algorithms for finding volatility of financial markets and monitoring the economy, performing the non-invasive evaluation of protein in ion channels, and prospecting elastic structures for residual stress, which can determine the aging or defects of materials and the elastic parameters of the Earth.Currently, there is a great need for better resolution in electrical impedance tomography. This project seeks improvements for this method using higher frequency data for the complete Maxwell system by its reduction to a vectorial Schroedinger equation and using complex and real geometrical optics solutions of this system. To make progress, the principal investigator uses anisotropic reductions to mainly upper-triangular systems and uses the second large parameter in scalar Carleman estimates. At present, there are few analytical results on Carleman estimates, describing the uniqueness and stability of anisotropic systems, including the challenging and important case of transversely isotropic elasticity. Inverse problems for complicated drift-diffusion systems, such as semiconductors and ion channels, abound with mathematical challenges including analytic conditions for uniqueness and reliable numerical algorithms. The principal investigator addresses these by using various asymptotic simplifications, dual problems, and methods similar to those used in inverse obstacle problems for scalar elliptic and parabolic equations.