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 ( K n ). In the diffusive scaling ( K n ≪ 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 K n → 0 . We tackle this problem by examining the stability of the inverse problem with varying K n . We show that the discrepancy in two measurements is amplified in the reconstructed parameters at the order of K n p ( p = 1 o r 2 ) , and as a result leads to ill-posedness in the zero limit of K n . Our results apply to both continuous and discrete settings. Lastly, some numerical tests are performed to validate these theoretical findings.