MOROZOV'S PRINCIPLE FOR THE AUGMENTED LAGRANGIAN METHOD APPLIED TO LINEAR INVERSE PROBLEMS

The augmented Lagragian method is an algorithm to compute saddle points for linearly constraint convex minimization problems. Recently it has received much attention, also under the name Bregman iteration, as an approach for regularizing inverse problems by applying the iteration to noisy data and stopping appropriately. Convergence and convergence rates have been shown for a priori stopping rules. This work shows convergence and convergence rates for this method when a special a posteriori rule, namely, Morozov's discrepancy principle, is chosen as a stopping criterion. Particularly, we treat the case in which this rule degenerates in the sense that the stopping indices do not tend to infinity as the noise level vanishes. Moreover, error estimates for the involved sequence of subgradients are pointed out. As potential fields of application we study implications of these results for particular examples in imaging. These are total-variation regularization as well as l(q) penalties with q is an element of [1, 2]. It is shown that Morozov's principle implies convergence and convergence rates for the iterates with respect to the metric of strict convergence and the l(q)-norm, respectively.