Forward-backward approximation of nonlinear semigroups in finite and infinite horizon

Andrés Contreras; Juan Peypouquet

doi:10.3934/cpaa.2021051

Article Contents

2021, Volume 20, Issue 5: 1893-1906. Doi: 10.3934/cpaa.2021051

This issue Previous Article Strict solutions to stochastic semilinear evolution equations in M-type 2 Banach spaces Next Article Dynamics of a 3D Brinkman-Forchheimer equation with infinite delay

Forward-backward approximation of nonlinear semigroups in finite and infinite horizon

Andrés Contreras^1,2,3, and
Juan Peypouquet^2,3,4, ,

1.
University of Paris-Saclay, CentraleSuplec, OPIS, Inria Saclay 91190, Gif-sur-Yvette, France
2.
Departamento de Ingeniería Matemática & , Centro de Modelamiento Matemático (CNRS IRL2807)
3.
FCFM, Universidad de Chile, Beauchef 851, Santiago, Chile
4.
Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, Bernoulliborg, Nijenborgh 9, 9747 AG Groningen, The Netherlands

^* Corresponding author: Juan Peypouquet
^* Corresponding author: Juan Peypouquet

Received: July 31, 2020

Revised: December 31, 2020

Early access: March 2021

Published: May 2021

Supported by FONDECYT Grant 1181179, CMM-Conicyt PIA AFB170001, ECOS-CONICYT Grant C18E04 and CONICYT-PFCHA/DOCTORADO NACIONAL/2016 21160994

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Abstract

This work is concerned with evolution equations and their forward-backward discretizations, and aims at building bridges between differential equations and variational analysis. Our first contribution is an estimation for the distance between iterates of sequences generated by forward-backward schemes, useful in the convergence and robustness analysis of iterative algorithms of widespread use in numerical optimization and variational inequalities. Our second contribution is the approximation, on a bounded time frame, of the solutions of evolution equations governed by accretive (monotone) operators with an additive structure, by trajectories constructed by interpolating forward-backward sequences. This provides a short, simple and self-contained proof of existence and regularity for such solutions; unifies and extends a number of classical results; and offers a guide for the development of numerical methods. Finally, our third contribution is a mathematical methodology that allows us to deduce the behavior, as the number of iterations tends to $ +\infty $, of sequences generated by forward-backward algorithms, based solely on the knowledge of the behavior, as time goes to $ +\infty $, of the solutions of differential inclusions, and viceversa.

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Mathematics Subject Classification: 34A60, 37L05, 65K15.

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