The density function reconstruction of surface sources from a single Cauchy measurement

The inverse problem of reconstructing sources is explored when a single boundary Cauchy data is postulated on the potential. We are particularly involved in sources supported by (hyper-)surfaces. Mild assumptions are required on the location of these supports and the calculation of the charge density function is then aimed. We consider a variational formulation, based on a duplication artifice of the potential and we check the symmetry and the positive definiteness of the weak problem. Because of the severe ill-posedness, the use of a regularization is mandatory for a safe approximation of the solution. Lavrentiev’s method is therefore recommended in the context owing to the symmetry and the positivity. We check why that regularization turns out to be a Tikhonov method for some underlying shadow equation that is not needed in computations and is therefore never explicitly constructed. Results stated in a wide literature for the Tikhonov regularization applies as well to our variational problem. An important consequence is that the Morozov Discrepancy Principle, we use for the selection of the regularization parameter yields a convergent strategy. Now, that the Discrepancy Principle requires the residual of that inaccessible ‘shadow equation’, we explain how the Kohn–Vogelius function allows for the computation of that residual.