Lagrangian methods for the regularization of discrete ill-posed problems

Abstract

In many science and engineering applications, the discretization of linear ill-posed problems gives rise to large ill-conditioned linear systems with right-hand side degraded by noise. The solution of such linear systems requires the solution of a minimization problem with one quadratic constraint depending on an estimate of the variance of the noise. This strategy is known as regularization. In this work, we propose to use Lagrangian methods for the solution of the noise constrained regularization problem. Moreover, we introduce a new method based on Lagrangian methods and the discrepancy principle. We present numerical results on numerous test problems, image restoration and medical imaging denoising. Our results indicate that the proposed strategies are effective and efficient in computing good regularized solutions of ill-conditioned linear systems as well as the corresponding regularization parameters. Therefore, the proposed methods are actually a promising approach to deal with ill-posed problems.