Instability estimates for the recovery of absorption in the diffusive regime of radiative transfer

TL;DR

This paper analyzes the stability transition in recovering absorption coefficients from radiative transfer data, revealing a shift from Hölder to logarithmic stability in the diffusive regime.

Contribution

It introduces a robust framework to study nonlinear stability phenomena in inverse radiative transfer problems, accommodating general geometries and critical stability transitions.

Findings

Identifies a transition from Hölder to logarithmic stability as the Knudsen number vanishes.

Develops a priori estimates for the radiative transfer equation based on previous strategies.

Handles general geometries by analyzing compression properties of the forward operator.

Abstract

We revisit the instability properties of the recovery of the absorption coefficient for the radiative transfer equation in the diffusive regime. To this end, we develop a rather robust framework building on [Koch-R\"uland-Salo, 2021] which allows us to deal with nonlinear critical stability transition phenomena. In particular, this permits us to consider rather general geometries based on the identification of compression properties of the forward operator. Given the albedo operator as the measurement data, we show that in the regime of vanishing Knudsen number there is a transition from H\"older to logarithmic stability in the inverse problem for the radiative transfer equation. As a central ingredient, we rely on suitable a priori estimates for the radiative transfer equation which we deduce by building on the strategy from [Dematt\`e-Vel\'azquez, 2025].