On the Transport Method for Hybrid Inverse Problems

Abstract

There are several hybrid inverse problems for equations of the form

$$\nabla \cdot D(x) \nabla u - \sigma (x) u = 0$$

in which we want to obtain the coefficients D and $\sigma$ on a domain $\varOmega$ 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 $u_2$. Bal and Ren have shown that for any nontrivial choices of $u_1$ and $u_2$, this method suffices to recover the coefficients almost everywhere on a dense set in $\varOmega$ Bal and Ren in (Inv Prob 075003 [3]). This article presents an alternate proof of the same result from a dynamical systems point of view.