An Inverse Source Problem in Radiative Transfer

Abstract

We consider the inverse source problem for the radiative transfer equation, under various assumptions on the scattering and absorption parameters, as well as on the accessible data. In such a setup, we measure the outgoing radiation intensity of the solution to the equation when an unknown source is present in the interior of the domain. The goal is to reconstruct the source from such measurements and obtain some form of stability estimate of the reconstruction on the data if possible. First, we extend the result of \cite{inversesource} to the case of partial data, where the absorption and scattering coefficients may lie in a certain dense open subset of $C^{2}(\overline{\Omega} \times \mathbb{S}^{n-1}) \times C^{2}(\overline{\Omega}, \mathbb{S}_{\theta'}^{n-1}; C^{n+1}(\mathbb{S}_{\theta}^{n-1}))$. Here it is shown one can recover sources supported in a particular subset of the domain, which we call the \textit{visible set}. We next show that for an open dense set of $C^{\infty}$ absorption and scattering coefficients, one can recover the part of the wave front set of the source that is supported in the \textit{microlocally visible set}, modulo a function in the Sobolev space $H^{k}$ for $k$ arbitrarily large. Following, we consider the case where the scattering kernel $k$ is small in suitable norm, and in this case we can reduce the smoothness requirements on $k$ from $C^{2}(\overline{\Omega} \times \mathbb{S}_{\theta'}^{n-1}; C^{n+1}(\mathbb{S}_{\theta}^{n-1}))$ to $C^{2}(\overline{\Omega} \times\mathbb{S}^{n-1} \times \mathbb{S}^{n-1})$. Finally, we demonstrate a numerical scheme based on the method in \cite{monard}, which we use to solve the inverse source problem in specific cases.