On the propagation of singularities for a class of linearised hybrid inverse problems

TL;DR

This paper analyzes how singularities propagate in linearised hybrid inverse problems in impedance tomography, identifying conditions under which singularities travel and demonstrating this through numerical experiments.

Contribution

It introduces a pseudo-differential framework to describe singularity propagation and links ellipticity loss to the movement of singularities in inverse problems.

Findings

Singularities propagate under certain non-elliptic conditions.

The directions of propagation are precisely characterized.

Numerical experiments visualize the propagation of singularities.

Abstract

For a general formulation of linearised hybrid inverse problems in impedance tomography, the qualitative properties of the solutions are analysed. Using an appropriate scalar pseudo-differential formulation, the problems are shown to permit propagating singularities under certain non-elliptic conditions, and the associated directions of propagation are precisely identified relative to the directions in which ellipticity is lost. The same result is found in the setting for the corresponding normal formulation of the scalar pseudo-differential equations. A numerical reconstruction procedure based of the least squares finite element method is derived, and a series of numerical experiments visualise exactly how the loss of ellipticity manifests itself as propagating singularities.