Approximation and uniqueness results for the nonlocal diffuse optical tomography problem

TL;DR

This paper establishes approximation and uniqueness results for the inverse problem of recovering diffusion and absorption coefficients in a nonlocal diffuse optical tomography model, using advanced mathematical techniques.

Contribution

It introduces new approximation and uniqueness theorems for the nonlocal diffuse optical tomography problem, extending known results to a nonlocal setting with innovative methods.

Findings

Solutions to local conductivity equations can be approximated by nonlocal solutions.

The nonlocal Dirichlet-to-Neumann map can approximate the classical one.

Absorption coefficient can be uniquely determined near the boundary under certain conditions.

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.