Determining two coefficients in diffuse optical tomography with incomplete and noisy Cauchy data

TL;DR

This paper addresses the challenging inverse problem of identifying conductivity and reaction coefficients in diffuse optical tomography using incomplete, noisy boundary data, proposing a regularization method with theoretical stability and convergence guarantees.

Contribution

It introduces a novel energy functional and total variation regularization approach for PDE-constrained optimization in diffuse optical tomography, with proven stability and convergence.

Findings

Proposed regularization method is stable under noisy data.

Finite element solutions converge in Lebesgue norms and Bregman distance.

Numerical case study supports theoretical results.

Abstract

In this paper we investigate the non-linear and ill-posed inverse problem of simultaneously identifying the conductivity and the reaction in diffuse optical tomography with noisy measurement data available on an accessible part of the boundary. We propose an energy functional method and the total variational regularization combining with the quadratic stabilizing term to formulate the identification problem to a PDEs constrained optimization problem. We show the stability of the proposed regularization method and the convergence of the finite element regularized solutions to the identification in the Lebesgue norms and in the sense of the Bregman distance with respect to the total variation semi-norm. To illustrate the theoretical results, a numerical case study is presented which supports our analytical findings.