Discrete inverse problem on grids

Abstract

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 Schr ¨odinger matrix and boundary value problems associated to it. 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 Schr ¨odinger matrix. The Schur complement plays an important role in matrix analysis, statistics, numerical analysis, and many other areas of mathematics and its applications.