Discrete inverse problems with internal functionals

TL;DR

This paper investigates the local uniqueness of resistor network inverse problems using internal power measurements, providing conditions under which the resistor configuration can be uniquely determined.

Contribution

It introduces a novel method inspired by continuum inverse problems to establish local uniqueness conditions for discrete resistor network inverse problems.

Findings

Derived local uniqueness conditions for resistor networks.

Extended the method to inverse Schrödinger problems and impedance cases.

Showed power measurements can be obtained from thermal noise currents.

Abstract

We study the problem of finding the resistors in a resistor network from measurements of the power dissipated by the resistors under different loads. We give sufficient conditions for local uniqueness, i.e. conditions that guarantee that the linearization of this non-linear inverse problem admits a unique solution. Our method is inspired by a method to study local uniqueness of inverse problems with internal functionals in the continuum, where the inverse problem is reformulated as a redundant system of differential equations. We use our method to derive local uniqueness conditions for other discrete inverse problems with internal functionals including a discrete analogue of the inverse Schr\"odinger problem and problems where the resistors are replaced by impedances and dissipated power at the zero and a positive frequency are available. Moreover, we show that the dissipated power measurements can be obtained from measurements of thermal noise induced currents.