Jorge P. Zubelli  

National Institute for Pure and Applied Mathematics (IMPA - Brazil)

Mathematical Methods for the Solution of Inverse Problems

Abstract:
Most of the problems that appear in Engineering and the Applied Sciences require the identification of parameters from observed data. This ubiquity lead to a tremendous interest on the field. In fact, the theory of Inverse Problems lies in the cross-roads of many fields including Analysis and Partial Differential Equations, Optimization, Statistics and Numerical Analysis. In this mini-course we shall concentrate on some aspects of Inverse Problems according to the following plan:

1. Inverse and Ill-posed problems: The tools of the trade. Singular Value Decomposition. Regularization. Noise modeling. Statistical Inverse Problems.
2. Classical Tomography: The Radon transfom. Analytical Results. Numerical Methods. Convergence and Stability of reconstructions.
3. Impedance Tomography: Some aspects of the forward problem. The Dirichlet to Neumann map. Calderon’s results. Uniqueness of the solutions and regularization. Optical diffusion tomography and infra-red imaging.
4. Inverse Problems in the the applied sciences: Examples from Mathematical Finance and Mathematical Biology.