Numerical Recovery of Source Singularities via the Radiative Transfer Equation with Partial Data

TL;DR

This paper demonstrates numerical methods to recover visible singularities of an internal source in the radiative transfer equation using partial data, employing a truncated Neumann series and providing theoretical justifications.

Contribution

The paper introduces a numerical scheme for approximating the normal operator in the inverse source problem with partial data, including theoretical analysis of the approximation's smoothness.

Findings

Effective detection of visible singularities in numerical simulations

Approximation of the normal operator with higher Sobolev regularity

Validation of the method's robustness with noisy data

Abstract

The inverse source problem for the radiative transfer equation is considered, with partial data. Here we demonstrate numerical computation of the normal operator $X_{V}^{*}X_{V}$ where $X_{V}$ is the partial data solution operator to the radiative transfer equation. The numerical scheme is based in part on a forward solver designed by F. Monard and G. Bal. We will see that one can detect quite well the visible singularities of an internal optical source $f$ for generic anisotropic $k$ and $\sigma$, with or without noise added to the accessible data $X_{V}f$. In particular, we use a truncated Neumann series to estimate $X_{V}$ and $X_{V}^{*}$, which provides a good approximation of $X_{V}^{*}X_{V}$ with an error of higher Sobolev regularity. This paper provides a visual demonstration of the authors' previous work in recovering the microlocally visible singularities of an unknown source from partial data. We also give the theoretical analysis necessary to justify the smoothness of the remainder when approximating the normal operator.