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

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

2010-10-25

arxiv.org/abs/1010.5181v1

Multiscale Model. Simul. 9 (2011) 1528-1548

Mathematics

Numerical Analysis

Scientific paper

10.1137/100812835

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.

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