Inverse problems and invisibility cloaking for FEM models and resistor networks

TL;DR

This paper explores inverse problems and invisibility cloaking in FEM models and resistor networks, revealing increased nonuniqueness in the discrete setting and demonstrating cloaking techniques for FEM models.

Contribution

It characterizes FEM models equivalent to resistor networks and analyzes nonuniqueness and cloaking in discrete inverse conductivity problems.

Findings

Discrete inverse problems exhibit greater nonuniqueness than continuous ones.

FEM models can be cloaked to mimic background media at the boundary.

Characterization of FEM models equivalent to resistor networks.

Abstract

In this paper we consider inverse problems for resistor networks and for models obtained via the Finite Element Method (FEM) for the conductivity equation. These correspond to discrete versions of the inverse conductivity problem of Calder\'on. We characterize FEM models corresponding to a given triangulation of the domain that are equivalent to certain resistor networks, and apply the results to study nonuniqueness of the discrete inverse problem. It turns out that the degree of nonuniqueness for the discrete problem is larger than the one for the partial differential equation. We also study invisibility cloaking for FEM models, and show how an arbitrary body can be surrounded with a layer so that the cloaked body has the same boundary measurements as a given background medium.