Recovering the conductances on grids

In this work, we present an algorithm to the recovery of the conductance of a n–dimensional grid. The algorithm is based in the solution of some overdetermined partial boundary value problems defined on the grid; that is, boundary value problem where the boundary conditions are set only in a part of the boundary (partial), and moreover in a fix subset of the boundary we prescribe both the value of the function and of its normal derivative (overdetermined). Our goal is to recover the conductance of a n–dimensional grid network with boundary using only boundary measurements and global equilibrium conditions. This problem is known as inverse boundary value problem. In general, inverse problems are exponentially ill–posed, since they are highly sensitive to changes in the boundary data. However, in this work we deal with a situation where the recovery of the conductance is feasible: grid networks. The recovery of the conductances of a grid network is performed here using its Schrödinger matrix and boundary value problems associated to it, which have been extensively studied in [2]. We will use some important results of this paper. Moreover, we use the Dirichlet–to–Robin matrix, also known as response matrix of the network, which contains the boundary information. It is a certain Schur complement of the Schrödinger matrix, and it was introduced in [1]. The Schur complement plays an important role in matrix analysis, statistics, numerical analysis, and many other areas of mathematics and its applications.