This paper is about a well-posed and well-conditioned inverse problem. Our particular problem is perhaps not realistic. Instead of measurements "at the boundary," we go right inside a network --all nodes and edges are treated equally. By probing the network we find its response to m different source terms. From that information about the system, the goal is to find the conductances on the m edges. The equations are highly nonlinear. What is attractive is the ap pearance of a convex potential Jimction, whose gradient is zero at the solution. The analysis of the problem becomes an analysis of this potential function.
Our problem is discrete rather than continuous (for now). The connections between nodes and edges of the network are given. They are recorded in an m by n incidence matrix A, which maps the potentials at the n nodes to the potential differences across the m edges. (The matrix A is a discrete analogue of a differential operator; it can be written down immediately for any network.) The conductances ej on the edges go into a positive diagonal matrix C. The system is governed by the symmetric positive-definite matrix ArCA. That is the fundamental matrix for the direct problem: conductances known, source terms known, and voltages and currents to be computed.