Stability of inverse transport equation in diffusion scaling and Fokker-Planck limit
For researchers in inverse problems and radiative transfer, this work quantifies stability degradation in two important asymptotic regimes, providing theoretical guidance for practical reconstruction.
The paper studies the stability of inverse transport equation in two scaling regimes: diffusive scaling and Fokker-Planck limit. It shows that stability degrades in the diffusive limit, and in the Fokker-Planck limit, full recovery of the scattering coefficient is less possible but a rescaled version is more practical.
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.