Convexification Numerical Method for a Coefficient Inverse Problem for the Riemannian Radiative Transfer Equation

TL;DR

This paper introduces the first globally convergent convexification numerical method for solving a coefficient inverse problem related to the Riemannian Radiative Transfer Equation, with proven convergence and numerical validation.

Contribution

It develops a novel convexification approach with Carleman Weight Function for RRTE, enabling global convergence in solving the inverse problem.

Findings

The method achieves global convergence for the inverse problem.

Numerical experiments confirm the theoretical convergence.

The approach effectively reconstructs the dielectric constant from data.

Abstract

The first globally convergent numerical method for a Coefficient Inverse Problem (CIP) for the Riemannian Radiative Transfer Equation (RRTE) is constructed. This is a version of the so-called \textquotedblleft convexification" method, which has been pursued by this research group for a number of years for some other CIPs for PDEs. Those PDEs are significantly different from the RRTE. The presence of the Carleman Weight Function (CWF) in the numerical scheme is the key element which insures the global convergence. Convergence analysis is presented along with the results of numerical experiments, which confirm the theory. RRTE governs the propagation of photons in the diffuse medium in the case when they propagate along geodesic lines between their collisions. Geodesic lines are generated by the spatially variable dielectric constant of the medium.