Some Inverse Problems for Obstacles and Drift-Diffusion and Elasticity Systems

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This project is concerned with analysis and design of numerical algorithms for reconstructing solutions and coefficients of partial differential equations from remote data. One of central topics is the study of increasing stability of the continuation of (acoustical and electromagnetic) waves. Another central topic is about uniqueness, stability, and methods of reconstruction of obstacles from boundary or scattering measurements, with emphasis on practical situations (in particular, on use of several frequencies or time data). The proposed research on drift-diffusion equations is concentrated around evaluation of doping profile in semiconductors and protein region in ion channels, as well finding volatility of option markets. The PI plans to obtain Carleman estimates for hyperbolic systems of anisotropic (including transversely isotropic) elasticity. All these research areas are challenging and central in applied mathematics. He will continue working on outstanding analytical problems, like uniqueness in the three-dimensional inverse conductivity problem with local data. Energy (including Carleman) estimates, harmonic and micro local analysis and potential theory will be main tools of this research. The research on increasing stability of continuation and of locating obstacles is crucial for enhancing resolution of remote sensing in a variety of civil and military applications. Finding the doped area of semiconductor devices is of value for quality control in computer industry, and monitoring of ion channels is of fundamental interest in biology and medicine. The study of nondestructive evaluation of elastic properties is quite important in aviation, car, and construction industries, geophysics, and medicine. The PI will train graduate students preparing them for challenges of science and industry. Goals of this project have important applications to engineering, geophysics, and medicine.