Morozov's principle for the augmented Lagrangian method applied to linear inverse problems

TL;DR

This paper establishes convergence rates for the augmented Lagrangian method in linear inverse problems when using Morozov's discrepancy principle as a stopping rule, with applications in imaging including total variation and ^q penalties.

Contribution

It provides the first convergence rate analysis for the augmented Lagrangian method with Morozov's principle in the context of imaging inverse problems.

Findings

Convergence rates are proven for the method with Morozov's principle.

Error estimates for subgradients are derived.

Results apply to total variation and ^q regularizations.

Abstract

The Augmented Lagrangian Method as an approach for regularizing inverse problems received much attention recently, e.g. under the name Bregman iteration in imaging. This work shows convergence (rates) for this method when Morozov's discrepancy principle is chosen as a stopping rule. Moreover, error estimates for the involved sequence of subgradients are pointed out. The paper studies implications of these results for particular examples motivated by applications in imaging. These include the total variation regularization as well as $\ell^q$ penalties with $q\in[1,2]$. It is shown that Morozov's principle implies convergence (rates) for the iterates with respect to the metric of strict convergence and the $\ell^q$-norm, respectively.