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.

Range-relaxed strategy applied to the Levenberg–Marquardt method with uniformly convex penalization term in Banach spaces

“…We then solve the problem (14) with Ω = Ω μ as in (17). According to last subsection, this can be performed by finding a pair ( x, λ) ∈ X × (0, ∞) satisfying…”

“…In the last few years, the range-relaxed strategy has been successfully adapted to linear problems in Banach spaces [15,16] and to the LM method for solving nonlinear problems in Hilbert spaces [7,13]. Further, some Kaczmarz variants have been studied [7,8] as well as inexact versions [17]. The LM method combined with the range-relaxed strategy in Banach spaces with more general penalization terms is still unavailable.…”