An Application of Source Inequalities for Convergence Rates of Tikhonov Regularization with a Non-differentiable Operator

TL;DR

This paper investigates convergence rates of Tikhonov regularization for solving ill-posed non-linear operator equations involving a non-differentiable operator, using source inequalities to establish linear convergence under various smoothness assumptions.

Contribution

It introduces a novel application of source inequalities to derive convergence rates for Tikhonov regularization with a non-differentiable operator related to image segmentation.

Findings

Established convergence rates under different smoothness conditions.

Proved the possibility of up to linear convergence in norm.

Applied variational inequalities to a non-differentiable operator context.

Abstract

In this paper we study Tikhonov regularization for the stable solution of an ill-posed non-linear operator equation. The operator we consider, which is related to an active contour model for image segmentation, is continuous, compact, but nowhere differentiable. Nevertheless we are able to derive convergence rates under different smoothness assumptions on the true solution by employing the method of variational or source inequalities. With this approach, we can prove up to linear convergence with respect to the norm.