Topology Reconstruction of a Resistor Network with Limited Boundary Measurements: An Optimization Approach

TL;DR

This paper presents a multistage optimization-based method for reconstructing the topology and edge resistances of a circular planar resistive network using limited boundary measurements.

Contribution

It introduces a novel multistage approach combining difference of convex programming and heuristic methods for topology and interior node reconstruction.

Findings

Successfully reconstructs network topology with limited measurements

Demonstrates the effectiveness of the optimization approach on a numerical example

Provides a framework for topology reconstruction in resistive networks

Abstract

A problem of reconstruction of the topology and the respective edge resistance values of an unknown circular planar passive resistive network using limitedly available resistance distance measurements is considered. We develop a multistage topology reconstruction method, assuming that the number of boundary and interior nodes, the maximum and minimum edge conductance, and the Kirchhoff index are known apriori. First, a maximal circular planar electrical network consisting of edges with resistors and switches is constructed; no interior nodes are considered. A sparse difference in convex program $\mathbf{\Pi}_1$ accompanied by round down algorithm is posed to determine the switch positions. The solution gives us a topology that is then utilized to develop a heuristic method to place the interior nodes. The heuristic method consists of reformulating $\mathbf{\Pi}_1$ as a difference of convex program $\mathbf{\Pi}_2$ with relaxed edge weight constraints and the quadratic cost. The interior node placement thus obtained may lead to a non-planar topology. We then use the modified Auslander, Parter, and Goldstein algorithm to obtain a set of planar network topologies and re-optimize the edge weights by solving $\mathbf{\Pi}_3$ for each topology. Optimization problems posed are difference of convex programming problem, as a consequence of constraints triangle inequality and the Kalmansons inequality. A numerical example is used to demonstrate the proposed method.