Approximation and uniqueness results for the nonlocal diffuse optical tomography problem

Abstract

We investigate the inverse problem of recovering the diffusion and absorption coefficients $(\sigma,q)$ in the nonlocal diffuse optical tomography equation $(-\text{div}( \sigma \nabla))^s u+q u =0 \text{ in }\Omega$ from the nonlocal Dirichlet-to-Neumann (DN) map $\Lambda^s_{\sigma,q}$. The purpose of this article is to establish the following approximation and uniqueness results. - Approximation: We show that solutions to the conductivity equation $ \text{div}( \sigma \nabla v)=0 \text{ in }\Omega$ can be approximated in $H^1(\Omega)$ by solutions to the nonlocal diffuse optical tomography equation and the DN map $\Lambda_\sigma$ related to conductivity equation can be approximated by the nonlocal DN map $\Lambda_{\sigma,q}^s$. - Local uniqueness: We prove that the absorption coefficient $q$ can be determined in a neighborhood $\mathcal{N}$ of the boundary $\partial\Omega$ provided $\sigma$ is already known in $\mathcal{N}$. - Global uniqueness: Under the same assumptions as for the local uniqueness result, and if one of the potentials vanishes in $\Omega$, then one can turn with the help of \ref{item 1 abstract} the local determination into a global uniqueness result. It is worth mentioning that the approximation result relies on the Caffarelli--Silvestre type extension technique and the geometric form of the Hahn--Banach theorem.