Quadratic rates of asymptotic regularity for the Tikhonov-Mann iteration

TL;DR

This paper establishes quadratic convergence rates for the Tikhonov-Mann iteration in W-hyperbolic spaces, extending nonlinear iteration analysis and connecting it to classical algorithms like Douglas-Rachfors and forward-backward methods.

Contribution

It provides the first quadratic rates of asymptotic regularity for the Tikhonov-Mann iteration in a nonlinear setting, unifying several algorithms under a common framework.

Findings

Quadratic convergence rates are established for the Tikhonov-Mann iteration.

The iteration generalizes and unifies Douglas-Rachfors and forward-backward algorithms with Tikhonov regularization.

Results extend convergence analysis from Hilbert spaces to W-hyperbolic spaces.

Abstract

In this paper, we compute quadratic rates of asymptotic regularity for the Tikhonov-Mann iteration in W-hyperbolic spaces. This iteration is an extension to a nonlinear setting of the modified Mann iteration defined recently by Bot, Csetnek and Meier in Hilbert spaces. Furthermore, we show that the Douglas-Rachfors and forward-backward algorithms with Tikhonov regularization terms are special cases, in Hilbert spaces, of our Tikhonov-Mann iteration.