Forward-backward approximation of evolution equations in finite and infinite horizon

TL;DR

This paper develops forward-backward discretization methods for evolution equations, providing convergence estimates, approximation techniques for solutions over finite horizons, and a methodology linking iterative algorithms with differential inclusions.

Contribution

It introduces new estimates for forward-backward schemes, offers a unified approach for approximating evolution equation solutions, and connects the long-term behavior of algorithms with differential inclusions.

Findings

Estimation of the distance between iterates in forward-backward schemes.

Approximation of evolution equation solutions over finite time frames.

A methodology linking iterative algorithms' behavior with differential inclusions.

Abstract

This research is concerned with evolution equations and their forward-backward discretizations. 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 variational analysis and optimization. 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 defined using 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.