Some inverse problems in elasticity, option pricing, semiconductors, and scattering theory.

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The project is concerned with four important areas of inverse problems for partial differential equations. We are looking for coefficients of systems of isotropic and anisotropic elasticity theory from dynamical boundary data in case of zero and nonzero initial conditions. We are concerned with uniqueness and stability of determination of coefficients. A related topic of general interest for inverse problems is about increased stability of the continuation and of the determination of the coefficients for Helmholtz type and parabolic equations when frequency or drift terms are large. Poor stability is one of major obstacles for applications of inverse problems, and better stability is very valuable. The main analytical tools will be Carleman estimates, harmonic and microlocal analysis, and potential theory. We plan to continue research on identification of the volatility coefficient of the famous Black- Scholes differential equation (of parabolic type) in important cases of volatility which is linear in time and in case of basket options. A new ingredient will be use of integral geometry. Finally, we consider identification of a conductivity coefficient (doping profile) in semiconductor models by using methods of inverse gravimetry and conductivity with few partial boundary measurements and of dual problems. One of goals is designing fast and reliable algorithms for numerical solution of these inverse problems.The work on elasticity aims to improve resolution of determination of elastic properties of the Earth, of human body, and of material used in industry (like steel) which are crucial for geophysics, medicine, and engineering. Research on inverse option pricing should help with monitoring and predicting financial markets in near future. Looking for doping profile is important for quality control in crucial area of manufacturing of computer chips.