# Stability of Stationary Inverse Transport Equation in Diffusion Scaling

## TL;DR

This paper investigates how the stability of reconstructing optical parameters from the stationary radiative transfer equation deteriorates as the Knudsen number approaches zero, revealing increased ill-posedness in the diffusive limit.

## Contribution

It provides a quantitative analysis of the stability loss in inverse transport problems as the system transitions to the diffusive regime, highlighting the challenges in parameter reconstruction.

## Key findings

- Discrepancy in measurements is amplified by a factor of $K_n^p$ in the reconstructed parameters.
- Inverse problem becomes increasingly ill-posed as $K_n o 0$.
- Theoretical results are validated with numerical tests.

## Abstract

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.