Convergence of variational regularization methods for imaging on Riemannian manifolds

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 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 state 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.