Inverse problems for the stationary transport equation in the diffusion scaling

TL;DR

This paper investigates the inverse problem for the stationary radiative transfer equation in the diffusion limit, revealing how stability deteriorates exponentially as the Knudsen number decreases, thus connecting it to the elliptic inverse problem.

Contribution

It derives stability estimates for the inverse RTE problem, showing the exponential worsening of stability as the Knudsen number approaches zero, linking it to the elliptic case.

Findings

Stability is Lipschitz across regimes but deteriorates exponentially as 1/Kn.

Inverse problem becomes severely ill-posed in the diffusion limit.

Numerical results confirm the theoretical stability deterioration.

Abstract

We consider the inverse problem of reconstructing the optical parameters of the radiative transfer equation (RTE) from boundary measurements in the diffusion limit. In the diffusive regime (the Knudsen number $\mathsf{Kn}\ll 1$), the forward problem for the stationary RTE is well approximated by an elliptic equation. However, the connection between the inverse problem for the RTE and the inverse problem for the elliptic equation has not been fully developed. This problem is particularly interesting because the former one is mildly ill-posed , with a Lipschitz type stability estimate, while the latter is well known to be severely ill-posed with a logarithmic type stability estimate. In this paper, we derive stability estimates for the inverse problem for RTE and examine its dependence on $\mathsf{Kn}$. We show that the stability is Lipschitz in all regimes, but the coefficient deteriorates as $e^{\frac{1}{\mathsf{Kn}}}$, making the inverse problem of RTE severely ill-posed when $\mathsf{Kn}$ is small. In this way we connect the two inverse problems. Numerical results agree with the analysis of worsening stability as the Knudsen number gets smaller.