Linearization of the inverse conductivity problem

TL;DR

This paper demonstrates that the inverse conductivity problem on graphs can be linearized by transforming the Dirichlet-to-Neumann map, simplifying the solution to a linear system, with potential extensions to continuous cases.

Contribution

It introduces a linearization technique for the inverse conductivity problem on graphs, reducing it to solving linear equations based on determinants of submatrices.

Findings

The map from log-conductivity to log-determinants is linear.

Solution reduces to solving linear systems from disjoint paths.

Algorithm potentially generalizes to planar, 3D, and continuous cases.

Abstract

A positive function (conductivity) on the edges of a graph induces the Dirichlet-to- Neumann map between boundary values of harmonic functions. The inverse conductivity problem is to find the conductivity from the Dirichlet-to-Neumann map. We will show that the map from logarithm of conductivity to the certain logarithms of the determinants of the submatrices of the Dirichlet-to-Neumann map is linear(!) and so the solution of the inverse problem is reduced to solution of the system of linear equations that arise from disjoint paths in the graph. We will make a calculation for a simple tensor product lattice graph and conjecture that it generalizes to planar and three dimensional graphs and also to the continuous case. Depending on the graph the algorithm resembles or not the layer-stripping.