When it is possible to input the governing equation(s), shape(s) and size(s) of the domain(s), boundary and initial conditions, material properties of the media contained in the field, and forces or sources, then the analysis determining the unknown field is considered mathematically well-posed, i.e. the solution exists, is unique and it depends continuously on the data. If any of these elements are unknown or unavailable, then the field problem becomes improperly defined (ill-posed) and is of an indirect (or inverse) type. The course will give an introduction to Inverse Problems. Various mathematical, numerical and statistical techniques for solving inverse problems will be described.
There is a background level of linear algebra, partial differential equations, numerical and functional analysis for which there are general courses. Also just enough physics to understand the phenomena of heat conduction, fluid flow, acoustics, optics and electromagnetism used to formulate the forward problems.
* Basic linear inverse problems - enough linear algebra and functional analysis to understand ill-conditioning and regularization of inverse problems.
* Basic techniques for linear inverse problems - truncated singular value decomposition, Tikhonov's regularization, parameter choice methods, etc.
* PDE theory for inverse problems - enough to read the main existence, uniqueness and stability papers, e.g. Isakov's book. Some mathematical techniques and concepts, e.g. Schauder fixed point theorem, contraction principle, Fredholm alternative, etc.
* Numerical methods for inverse problems including FEM and BEM for forward problem solution and iterative regularization methods. Level set method. Constrained minimization gradient based methods. Possibly Bayesian approach and introduction to MCMC.