# Applications of Kinetic Tools to Inverse Transport Problems

## TL;DR

This paper develops a unified PDE-based framework using classical tools like energy estimates and averaging lemmas to solve inverse problems for kinetic equations, including nonlinear cases and lower-dimensional domains.

## Contribution

It introduces a novel PDE analysis approach for inverse kinetic problems, covering nonlinear and 2D cases, expanding the scope of existing methods.

## Key findings

- Unified framework for reconstructing absorption coefficients
- First nonlinear inverse transport result in this context
- Extension of scattering coefficient recovery from 3D to 2D domains

## Abstract

We show that the inverse problems for a class of kinetic equations can be solved by classical tools in PDE analysis including energy estimates and the celebrated averaging lemma. Using these tools, we give a unified framework for the reconstruction of the absorption coefficient for transport equations in the subcritical and critical regimes. Moreover, we apply this framework to obtain, to the best of our knowledge, the first result in a nonlinear setting. We also extend the result of recovering the scattering coefficient in [14] from 3D to 2D convex domains.