Analysis and Regularization of Problems in Diffuse Optical Tomography

Abstract

In this paper we consider the regularization of the inverse problem of diffuse optical tomography by standard regularization methods with quadratic penalty terms. We therefore investigate in detail the properties of the associated forward operators and derive continuity and differentiability results, which are based on derivation of $W^{1,p}$ regularity of solutions for the governing elliptic boundary value problems. We then show that Tikhonov regularization can be applied for a stable solution and that the standard convergence and convergence rates results hold. Our analysis also allows us to ensure convergence of iterative regularization methods, which are important from a practical point of view.