STATISTICAL METHODOLOGY FOR INVERSE PROBLEMS

Because, in the solution of practical inverse problems, the goal is to recover from available observational data (indirect measurements) as much information as possible about the underlying phenomenon, statistical methodologies represent a natural choice for the construction of algorithms. For example, Bayes' theorem and the EM-Methodology are widely used. In addition, stochastic modeling has played a key role when the underlying phenomenon is known to involve randomness, such as in radioactive decay. Such situations arise in nuclear medical imaging. Independently, it is known from theoretical numerical analysis that algorithms for inverse problems, in order to be useful, must in some appropriate way control the inherent improperly posedness. The purpose of this paper is to analyze the nature and extent of the stabilization induced by different statistical methodologies for inverse problems.