Convergence of Variational Regularization Methods for Imaging on Riemannian Manifolds

TL;DR

This paper investigates the convergence properties of variational regularization methods, specifically Tikhonov regularization, for solving ill-posed operator equations on finite-dimensional Riemannian manifolds, with theoretical analysis and numerical experiments.

Contribution

It extends the analysis of variational regularization methods to the setting of Riemannian manifolds, including well-posedness, stability, convergence, and numerical implementation.

Findings

Proved well-posedness and convergence of regularization methods on manifolds.

Established convergence rates for the regularization techniques.

Demonstrated effectiveness through numerical experiments on inverse problems.

Abstract

We consider abstract operator equations $Fu=y$, where $F$ is a compact linear operator between Hilbert spaces $U$ and $V$, which are function spaces on \emph{closed, finite dimensional Riemannian manifolds}, respectively. This setting is of interest in numerous applications such as Computer Vision and non-destructive evaluation. In this work, we study the approximation of the solution of the ill-posed operator equation with Tikhonov type regularization methods. We prove well-posedness, stability, convergence, and convergence rates of the regularization methods. Moreover, we study in detail the numerical analysis and the numerical implementation. Finally, we provide for three different inverse problems numerical experiments.