Implicit error bounds for Picard iterations on Hilbert spaces

Journal article (2018)

Authors

D. Russell Luke University of Göttingen
Hieu Thao Nguyen Team Raf Van de Plas - Mechanical, Maritime and Materials Engineering
M.K. Tam University of Göttingen
Research Group
Team Raf Van de Plas (Mechanical, Maritime and Materials Engineering) (TU Delft)
Averaged operators Fixed points Metric regularity Metric subregularity Picard iteration Error bounds Nonexpansiveness Strong convergence
More Info
expand_more

Published Date

2018

Language

English

Reuse Rights

Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons.

Faculty

Mechanical, Maritime and Materials Engineering

Department

Delft Center for Systems and Control

Research Group

Team Raf Van de Plas

Abstract

We investigate the role of error bounds, or metric subregularity, in the convergence of Picard iterations of nonexpansive maps in Hilbert spaces. Our main results show, on one hand, that the existence of an error bound is sufficient for strong convergence and, on the other hand, that an error bound exists on bounded sets for nonexpansive mappings possessing a fixed point whenever the space is finite dimensional. In the Hilbert space setting, we show that a monotonicity property of the distances of the Picard iterations is all that is needed to guarantee the existence of an error bound. The same monotonicity assumption turns out also to guarantee that the distance of Picard iterates to the fixed point set converges to zero. Our results provide a quantitative characterization of strong convergence as well as new criteria for when strong, as opposed to just weak, convergence holds.