We will use these models to illustrate some practical situations, in which may appear more complex models, leading to the solution of inverse problems of great importance in medical and engineering applications. Such problems are those which appear, for example, in Electrical Tomography, the study of thermophysical characteristics in processes of heat conduction, in the Inverse Electroencephalography and the Inverse Electrocardiography.
In all these problems the goal is to identify sources (f) or coefficients (ε) from additional measurements of the potential u on the boundary of the region Ω, identify the flow in one part of the boundary from knowledge of the potential in other part of it, identify unknown parts of the boundary of Ω which are not reachable from the unbounded connected component of the complement of Ω when these parts are the boundary of some inclusion with ideal insulating or conducting properties .
These problems fall into the category of ill-posed inverse problems whose solution is very sensitive to measurement errors of the data. In some of them there are theoretical results on existence and uniqueness of the identification problem when it is assumed that the data are measured without error. In the realistic case in which the data are considered given with some error are needed tools of regularization theory to obtain numerically stable solutions of the respective identification problems.
However, these results are not applicable in practice because in general require an infinite amount of measurements to make the identification which is not available. That is why in practice it is necessary to require some kind of "prior information" about the term to identify so that it can be determined approximately from a finite number of measurements given with error.
In the talk we consider an important case for applications, when sources or coefficients to be determined are considered piecewise constant.
It is also known that in the solution of inverse problems, the discretization is an additional source of ill- posedness. In the talk will discuss the importance of making appropriate discretizations of the models "consistent" with the measurement errors.