Ibrahima Dione | Searcharxiv

arXiv2025

Stabilization of the Gradient Method for Solving Linear Algebraic Systems -- A Method Related to the Normal Equation

Although it is relatively easy to apply, the gradient method often displays a disappointingly slow rate of convergence. Its convergence is specially based on the structure of the matrix of the algebraic linear system, and on the choice of the stepsize defining the new iteration. We propose here a simple and robust stabilization of the gradient method, which no longer assumes a structure on the matrix (neither symmetric, nor positive definite) to converge, and which no longer requires an approach on the choice of the stepsize. We establish the global convergence of the proposed stabilized algorithm under the only assumption of nonsingular matrix. Several numerical examples illustrating its performances are presented, where we have tested small and large scale linear systems, with and not structured matrices, and with well and ill conditioned matrices.

math.NA↗

arXiv2024

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

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.

math.NA↗