Numerical methods for parameter identification in stationary radiative transfer

TL;DR

This paper develops a Tikhonov regularization approach in Banach spaces for stable parameter identification in stationary radiative transfer, establishing theoretical properties and providing numerical validation.

Contribution

It introduces a novel regularization framework using optimal control constrained by integro PDEs for radiative transfer parameter identification.

Findings

Existence of minimizers proven.

Differentiability properties established for numerical algorithms.

Numerical results support theoretical analysis.

Abstract

We consider the identification of scattering and absorption rates in the stationary radiative transfer equation. For a stable solution of this parameter identification problem, we consider Tikhonov regularization within Banach spaces. A regularized solution is then defined via an optimal control problem constrained by an integro partial differential equation. By establishing the weak-continuity of the parameter-to-solution map, we are able to ensure the existence of minimizers and thus the well-posedness of the regularization method. In addition, we prove certain differentiability properties, which allow us to construct numerical algorithms for finding the minimizers and to analyze their convergence. Numerical results are presented to support the theoretical findings and illustrate the necessity of the assumptions made in the analysis.