Recovering the conductances on grids: A theoretical justification

Abstract

In this work, we present an overview of the work developed by the authors in the context of inverse problems on nite networks. This study performs an extension of the pioneer studies by E.B. Curtis and J.A. Morrow, and sets the theoretical basis for solving inverse problems on networks. We present just a glance of what we call overdetermined partial boundary value problems, in which any data are not prescribed on a part of the boundary, whereas in another part of the boundary both the values of the function and of its normal derivative are given. The resolvent kernels associated with these problems are described and they are the fundamental tool to perform an algorithm for the recovery of the conductance of a 3{dimensional grid. We strongly believe that the columns of the partial overdetermined Poisson kernel are the discrete counterpart of the so{called CGO solutions (complex geometrical optic solutions) that, in their turn, are the key to solve inverse continuous problems on planar domains. Finally, we display the steps needed to recover the conductances in a 3{dimensional grid.