Some Theoretical and Applied Inverse Problems

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This project is dedicated to the study of uniqueness and stability insome important inverse problems, in particular to identification of diffusion coefficientsand the speed of propagation for scalar partial differential equations and (Maxwell andelasticity)systems. Tools will be new Carleman type estimates andinteraction between control theory and inverse problems. Also potentialtheory is to be utilized in the further study of inverse gravimetry andvorticity problems. In case of many boundary measurements the effort will bemade to resolve the fundamental uniqueness question in the inverseconductivity problem with the data on a part of the boundary by usinglocalized solutions of elliptic equations recently constructed by Greenleafand Uhlmann. The PI will try to find cases of increased stability in inverseproblems, in particular obtaining Lipschitz conditional stability estimates forrecovery of nonlinear (space independent) elliptic and parabolic equationsfrom boundary measurements and quantifying increased stability inprospecting by stationary waves with higher frequency. Applications of theoretical results will be in identification of volatility in options markets. We expect to develop a very efficient and reliable algorithm based on the modeling of volatility. This problem is of fundamental importance for prediction of options markets and evaluation of economical stability from current market data. Another expected application is recovery of the fundamental physical relations (in chemical equations, heat conduction) from experimental data. This problem is of growing importance for contemporary engineering due to the discovery of new materials and the use of high temperatures.The theory of inverse gravimetry combined with available models of vorticity is expected to achieve progress in evaluation of turbulence (e.g. past aircrafts) from distant measurements of pressure. This problem has important implications for increasing capability of large airports. The project will involve graduate students and stimulate their interest in mathematical problems of practical importance.