On the solvability of the discrete conductivity and Schr\"odinger inverse problems

TL;DR

This paper investigates the uniqueness and solvability of inverse boundary value problems on graphs, demonstrating that under certain conditions, the edge weights or potentials are uniquely determined by boundary measurements, with exceptions on a measure-zero set.

Contribution

It introduces a discrete complex geometric optics approach to establish generic solvability and uniqueness for inverse problems on graphs, extending previous continuous methods.

Findings

Linearized inverse problems are solvable for almost all conductivities or potentials.

Unique determination of conductivities or potentials from boundary data holds outside a zero measure set.

The approach applies a discrete analogue of complex geometric optics to graph inverse problems.

Abstract

We study the uniqueness question for two inverse problems on graphs. Both problems consist in finding (possibly complex) edge or nodal based quantities from boundary measurements of solutions to the Dirichlet problem associated with a weighted graph Laplacian plus a diagonal perturbation. The weights can be thought of as a discrete conductivity and the diagonal perturbation as a discrete Schr\"odinger potential. We use a discrete analogue to the complex geometric optics approach to show that if the linearized problem is solvable about some conductivity (or Schr\"odinger potential) then the linearized problem is solvable for almost all conductivities (or Schr\"odinger potentials) in a suitable set. We show that the conductivities (or Schr\"odinger potentials) in a certain set are determined uniquely by boundary data, except on a zero measure set. This criterion for solvability is used in a statistical study of graphs where the conductivity or Schr\"odinger inverse problem is solvable.