| Abstract: |
| We consider the inverse problem of reconstructing the optical parameters for stationary radiative transfer equation (RTE) from velocity-averaged measurements. The RTE describes the dynamics of the distribution for photon particles. It often contains multiple scales characterized by the magnitude of a dimensionless parameter -- the Knudsen number ($\text{Kn}$). In the diffusive scaling ($\text{Kn} \ll 1$), 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 inverse transport equation, which raises the question of how uniqueness being lost as $\text{Kn} \to 0$. We tackle this problem by examining the stability of inverse problem with varying $\text{Kn}$. We show that, the discrepancy in two measurements is amplified in the reconstructed parameters at the order of $\text{Kn}^{-p}$ ($p = 1$ or $2$), and as a result lead to ill-posedness in the zero limit of $\text{Kn}$. Our results apply to both continuous and discrete settings. Some numerical tests are performed in the end to validate these theoretical findings. |
|