On the data completion problem for Laplace's equation

The purpose of this paper is the study and the resolution of the inverse problem for the Laplace equation, including the case of data completion problem where it is to cover the missing data on the inaccessible part of the boundary of a domain from measurements on the accessible part. Furthermore, we present a survey of the inverse problem of reconstructing the missing data for the Laplace equation. We describe the notion of ill-posed problems; namely, the results concerning the existence, uniqueness and stability of their solutions. In addition, we present several areas and fields of applications of this kind of problem. We also include the different developed methods for solving this problem, discussing their advantages and inconveniences. Numerical results with the iterative KMF algorithm and the developed variant are presented.