Regularization of Linear Ill-posed Problems by the Augmented Lagrangian Method and Variational Inequalities

TL;DR

This paper extends convergence rate results for the Augmented Lagrangian Method applied to linear ill-posed problems, using variational inequalities to handle lower order rates and different distance measures, including applications to sparsity regularization.

Contribution

It introduces a novel approach using variational inequalities to derive convergence rates for the Augmented Lagrangian Method beyond classical assumptions.

Findings

Convergence rates established under variational inequalities.

Extension to different distance measures beyond Bregman distance.

Application to sparsity-promoting regularization with restricted injectivity.

Abstract

We study the application of the Augmented Lagrangian Method to the solution of linear ill-posed problems. Previously, linear convergence rates with respect to the Bregman distance have been derived under the classical assumption of a standard source condition. Using the method of variational inequalities, we extend these results in this paper to convergence rates of lower order, both for the case of an a priori parameter choice and an a posteriori choice based on Morozov's discrepancy principle. In addition, our approach allows the derivation of convergence rates with respect to distance measures different from the Bregman distance. As a particular application, we consider sparsity promoting regularization, where we derive a range of convergence rates with respect to the norm under the assumption of restricted injectivity in conjunction with generalized source conditions of H\"older type.