Tikhonov-like methods with inexact minimization for solving linear ill-posed problems

Abstract

In this article we propose and study the properties of three distinct algorithms for obtaining stable approximate solutions for systems of ill-posed equations, modeled by linear operators acting between Hilbert spaces. Based on Tikhonov-like methods with uniformly convex penalty terms, we develop new versions with inexact minimization of both one step and iterated-Tikhonov methods. For the case of one step methods, we propose two distinct algorithms, one based in a priori and one based in a posteriori choice of the regularization parameter. Convergence and stability properties are provided, as well as optimal convergence rates (under appropriate source conditions). The third algorithm is based in a variant of the iterated-Tikhonov method with a posteriori choice of the sequence of penalization parameters. For this algorithm, we prove stability for noisy data and the regularization property.