Stability of Stationary Inverse Transport Equation in Diffusion Scaling
This work provides a theoretical understanding of how ill-posedness emerges in inverse transport problems as the diffusion limit is approached, which is relevant for researchers in inverse problems and radiative transfer.
The paper studies the stability of reconstructing optical parameters from velocity-averaged measurements in the stationary radiative transfer equation under diffusive scaling, showing that the discrepancy in measurements is amplified by a factor of Kn^p (p=1 or 2) as the Knudsen number Kn approaches zero, leading to ill-posedness. Numerical tests validate the theoretical results.
We consider the inverse problem of reconstructing the optical parameters for 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 ($K_n \ll 1$), 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 $K_n \rightarrow 0$. We tackle this problem by examining the stability of inverse problem with varying $K_n$. We show that, the discrepancy in two measurements is amplified in the reconstructed parameters at the order of $K_n^p~ (p = 1\text{ or} ~2)$, and as a result lead to ill-posedness in the zero limit of $K_n$. Our results apply to both continuous and discrete settings. Some numerical tests are performed in the end to validate these theoretical findings.