Stability of inverse transport equation in diffusion scaling and Fokker-Planck limit

TL;DR

This paper investigates the stability of inverse problems for the radiative transfer equation under diffusive and Fokker-Planck scalings, revealing how stability degrades or changes in these limits, impacting reconstruction accuracy.

Contribution

It analyzes the stability of inverse radiative transfer problems in diffusive and Fokker-Planck regimes, showing how limits affect reconstruction feasibility and stability.

Findings

Stability degrades in the diffusive limit.

Full recovery of scattering coefficients becomes less feasible in the Fokker-Planck limit.

Rescaled scattering coefficients are more practical to recover in the Fokker-Planck regime.

Abstract

We consider the inverse problem of reconstructing the scattering and absorption coefficients using boundary measurements for a time dependent radiative transfer equation (RTE). As the measurement is mostly polluted by errors, both experimental and computational, an important question is to quantify how the error is amplified in the process of reconstruction. In the forward setting, the solution to the RTE behaves differently in different regimes, and the stability of the inverse problem vary accordingly. In particular, we consider two scalings in this paper. The first one concerns with a diffusive scaling whose macroscopic limit is a diffusion equation. In this case, we showed, following the similar approach as in [Chen, Li and Wang, arXiv:1703.00097], that the stability degrades when the limit is taken. The second one considers a highly forward peaked scattering, wherein the scattering operator is approximated by a Fokker-Planck operator as a limit. In this case, we showed that a fully recover of the scattering coefficient is less possible in the limit, whereas obtaining a rescaled version of the scattering coefficient becomes more practice friendly.