On July 6, 2022 at 16:00 EEST, Pedro Pinto (Technische Universität Darmstadt) will give a talk in the Proof Mining Seminar.
Title: The alternating Halpern-Mann iteration
The Halpern and Krasnoselskii-Mann iterations are two important methods for approximating zeros of nonexpansive maps. Several proof mining results have focused on the study on these iterative schemas and variants thereof. Recently , together with Bruno Dinis, the alternating Halpern-Mann iteration (HM) was introduced for approximating common fixed points of two nonexpansive maps. As the name suggests, this method alternates between the Halpern and Krasnoselskii-Mann definitions. Under some conditions, it was possible to establish asymptotic regularity and strong convergence towards a common fixed point, in the setting of CAT(0) spaces. In this talk, we will look at the motivations behind this iteration, and discuss how such a strong convergence result was made possible by techniques and ideas from the proof mining program (). In recent work , together with Laurențiu Leuștean, it was possible to generalize the asymptotic regularity result to UCW-hyperbolic spaces. We will discuss how this suggests that HM is a proper mixing of the Halpern and Mann iterations.
 B. Dinis, and P. Pinto. Strong convergence for the alternating Halpern-Mann iteration in CAT(0) spaces. arXiv:2112.14525, Submitted, 2021.
 F. Ferreira, L. Leuștean, and P. Pinto. On the removal of weak compactness arguments in proof mining. Advances in Mathematics, 354:106728, 2019.
 L. Leuștean, and P. Pinto. Rates of asymptotic regularity for the alternating Halpern-Mann iteration. arXiv:2206.02226, Submitted, 2022.
 U. Kohlenbach, and L. Leuștean. Effective metastability of Halpern iterates in CAT(0) spaces. Advances in Mathematics, 231(5):2526-2556, 2012.
 H. Cheval, and L. Leuștean. Quadratic rates of asymptotic regularity for the Tikhonov-Mann iteration. Optimization Methods and Software, 2022.
 H. Cheval, U. Kohlenbach, and L. Leuștean. On modified Halpern and Tikhonov-Mann iterations. arXiv:2203.11003, Submitted, 2022.
Google Meet link: https://meet.google.com/xwm-syvx-bbr