The PI intends to continue his study of uniqueness and stability for certain important inverse problems. The contemporary methods of the theory of partial differential equations will be used. Those include Carleman type estimates, the Fourier integral operators, potential theory, and general functional analytic approach. An emphasis is on inverse problems with local/partial boundary measurements, on increasing stability by isolating ill-posedness in a simple linear transform, on recovery of non-linear equations, and on interaction between inverse problems and optimal control theory. Based on the theory the PI plans to develop efficient numerical algorithms for the following important applied inverse problems. The applied focus of the project will be on - the problem of electromagnetic detection of the active part of human brain - identifying a source of acoustical noise located on the surface of an aircraft cabin from interior measurements - finding constitutive laws for complicated non-linear physical systems, including underground flows in porous media and chemical reactions - reconstruction of the volatility coefficient of options trading from current market data. The second problem is of immediate importance in designing less noisy cars and midsize aircraft. The fourth one is about prediction of the future state of the very important financial area influencing the life of contemporary society.
