Stability of stationary inverse transport equation in diffusion scaling

We consider the inverse problem of reconstructing the optical parameters for the stationary radiative transfer equation (RTE) from velocity-averaged measurement. The RTE often contains multiple scales, characterized by the magnitude of a dimensionless parameter-the Knudsen number (Kn). In the diffusive scaling (Kn ≪ 1), the stationary RTE is well approximated by an elliptic equation in the forward setting. However, the inverse problem for the elliptic equation is acknowledged to be severely ill-posed, as compared to the well-posedness of the inverse transport equation, which raises the question of how uniqueness is lost as Kn → 0. We tackle this problem by examining the stability of the inverse problem with varying Kn. We show that the discrepancy in two measurements is amplified in the reconstructed parameters at the order of Kn^p ( p = 1 or 2), and as a result leads to ill-posedness in the zero limit of Kn. Our results apply to both continuous and discrete settings. Lastly, some numerical tests are performed to validate these theoretical findings.