A strongly convergent Krasnosel'ski\v{\i}-Mann-type algorithm for finding a common fixed point of a countably infinite family of nonexpansive operators in Hilbert spaces

TL;DR

This paper introduces a new strongly convergent Krasnosel'ski-mann-type algorithm for finding common fixed points of infinitely many nonexpansive operators in Hilbert spaces, with applications to variational image reconstruction.

Contribution

It develops a novel algorithm with proven strong convergence for infinite families of nonexpansive operators, extending existing methods.

Findings

The algorithm converges strongly to the fixed point with minimum norm.

Variable step size forward-backward method outperforms constant step size in experiments.

Effective in solving variational and split feasibility problems in infinite-dimensional spaces.

Abstract

In this article, we propose a Krasnosel'ski\v{\i}-Mann-type algorithm for finding a common fixed point of a countably infinite family of nonexpansive operators $(T_n)_{n \geq 0}$ in Hilbert spaces. We formulate an asymptotic property which the family $(T_n)_{n \geq 0}$ has to fulfill such that the sequence generated by the algorithm converges strongly to the element in $\bigcap_{n \geq 0} \operatorname{Fix} T_n$ with minimum norm. Based on this, we derive a forward-backward algorithm that allows variable step sizes and generates a sequence of iterates that converge strongly to the zero with minimum norm of the sum of a maximally monotone operator and a cocoercive one. We demonstrate the superiority of the forward-backward algorithm with variable step sizes over the one with constant step size by means of numerical experiments on variational image reconstruction and split feasibility problems in infinite dimensional Hilbert spaces.