A Note on the Transport Method for Hybrid Inverse Problems

TL;DR

This paper provides an alternative proof, from a dynamical systems perspective, that the transport method can recover coefficients in certain hybrid inverse problems using two solutions.

Contribution

It offers a new proof approach for the transport method's effectiveness in recovering coefficients, complementing existing arguments.

Findings

Transport method recovers coefficients on a dense set in the domain.

Alternative proof from dynamical systems perspective.

Method applies to equations with solutions known on the domain.

Abstract

There are several hybrid inverse problems for equations of the form $\nabla \cdot D \nabla u - \sigma u = 0$ in which we want to obtain the coefficients $D$ and $\sigma$ on a domain $\Omega$ when the solutions $u$ are known. One approach is to use two solutions $u_1$ and $u_2$ to obtain a transport equation for the coefficient $D$, and then solve this equation inward from the boundary along the integral curves of a vector field $X$ defined by $u_1$ and $u_2$. It follows from an argument of Guillaume Bal and Kui Ren that for any nontrivial choices of $u_1$ and $u_2$, this method suffices to recover the coefficients on a dense set in $\Omega$. This short note presents an alternate proof of the same result from a dynamical systems point of view.