Inverse problems in partial differential equations and related topics (potential theory, uniqueness of the
continuation and Carleman estimates, nonlinear functional analysis and calculus of variations). I have a particular
interest and results in the following inverse problems:
- Inverse problem of gravimetry (general uniqueness conditions and local solvability theorems) and related
problems of imaging including prospecting active part of the brain and the source of noise of the aircraft from
exterior measurements of electromagnetic and acoustical fields.
- Inverse problem of conductivity (uniqueness of discontinuous conductivity and numerical methods) and their
applications to medical imaging and nondestructive testing of materials for cracks and inclusions.
- Inverse scattering problem (uniqueness and stability of penetrable and soft scatterers).
- Finding constitutional laws from experimental data (reconstructing nonlinear partial differential equation from all
or some boundary data).
- Uniqueness of the continuation for hyperbolic equations and systems of mathematical physics. This is quite
challenging and important (for optimal control and inverse problems) area, and a sophisticated mathematical
techniques (including Carleman estimates, pseudo convexity, and microlocal analysis) is used here.
- The inverse option pricing problem. Recovery of volatility from current option prices.
This fundamental inverse financial problem has links to inverse parabolic problems with final overdetermination.
It is well-posed, but difficult due to nonlinearity.