Regularization of Discrete Ill-Conditioned Problems Done Right -- I

TL;DR

This paper introduces a stabilized regularization method for solving rank-deficient and ill-posed problems, ensuring minimal norm solutions and improved stability regardless of the regularization norm used.

Contribution

It proposes a norm-independent stabilization technique that guarantees minimal norm solutions and enhances stability in regularization of ill-posed problems.

Findings

Provides minimal norm solutions for unperturbed problems.

Achieves more accurate and stable solutions with over-regularization.

Validated on standard ill-posed numerical examples.

Abstract

When solving rank-deficient or discrete ill-posed problems by regularization methods, the choice of the regularization parameter is crucial. It is also of interest, the regularization norm used in the selection of the solution. In this work, we propose a stabilization of the existing regularization methods to address the delicate task of choosing this parameter. The analysis we carried out is independent of the chosen regularization norm. Under an unperturbed data least squares problem and of a maximal rank matrix, the stabilized-regularized method we propose provides the minimal norm solution whatever the chosen positive regularization parameter. And under a perturbed data least squares problem, this approach provides increasingly accurate and stable approximations of the minimal norm solution with respect to a refined mesh and a huge regularization parameter (over-regularization). We also investigate standard rank-deficient and ill-posed numerical examples corroborating the theoretical analysis, where the accuracy and the stability of the proposed approach is widely discussed.