Abstract:
The radiative transport equation (RTE) is a common model to describe the propagation of particles (e.g., photons) in some underlying medium or tissue while taking into account scattering and absorption affects via according parameters. In my thesis I concentrate on the inverse transport problem of the RTE where the boundary measurements are used to reconstruct the scatter and absorption coefficients. According to Choulli and Stefanov this inverse problem is well defined in dimensions n>2 (under some additional requirements). It is known by virtue of Bensoussan, Lions and Papanicolaou that in the high scattering regime (Kn->0 for Kn the Knudsen number) and under additional smoothness assumptions on the parameters and the boundary of the domain the RTE can be approximated by a diffusion equation; the constraints under which this approximation is valid are subsumed under the notion diffusion regime. It is then an interesting question how the stability of the inverse transport problem behaves in the diffusion regime; inverse problems to diffusion equations (e.g., the Caledrón problem or the backwards heat equation) are known to be severely ill posed. In my thesis I combine results of Lai, Li and Uhlmann as well as Zhao and Zhong (both from 2019) that show that the inverse problem of the stationary RTE in the diffusion regime enjoys logarithmic stability as well as exponential instability. In my talk I will mainly focus on the instability statements from Zhao and Zhong. I will outline how they can be obtained by utilizing a general framework for instability considerations from Mandache's work on the instability of the Calderón problem.