Volume V - Ill-posed problems and regularization theory

"There are people that, if you told them a result, are able to evolve from their own inner consciousness what the steps were that led to that result." These words of Scherlock Holmes in "A Study of Scarlet" of Sir Conan Doyle capture much of the essence of scientists' everyday work. Science deals constantly with inverse problems. Phenomena happen because causes produce them. The scientist measures the effects of a phenomenon and try to reason from the data backward to the causes of that phenomenon. However inverse problems are frequently ill-posed, in the sense that the solution to the problem, if exists unique, does not depend continuosly on the data. This instability requires introducing mathematical regularization techniques. The physical ill-posed problems considered in this series of papers regard primarily the inverse problem of the scattering theory, that is, the problem of recovering the unknown scattering potential from the scattered data, and the inverse optical imaging problem. The main mathematical subjects involved in these problems are the theory of the analytic continuation and the Fredholm integral equations of the first kind. Deterministic Tikhonov-like regularization, probabilistic methods and procedures based on the topological-metric information theory are the tools developed and implemented in these papers for regularizing ill-posed problems.