Inducing strong convergence of trajectories in dynamical systems associated to monotone inclusions with composite structure

TL;DR

This paper develops and analyzes dynamical systems with regularization techniques to ensure strong convergence to minimum norm solutions for monotone inclusion problems, with theoretical proofs and numerical comparisons.

Contribution

It introduces regularized dynamical systems of Krasnoselskii-Mann, forward-backward, and forward-backward-forward types that guarantee strong convergence to minimum norm solutions.

Findings

Proves strong convergence of trajectories with Tikhonov regularization.

Demonstrates qualitative differences between regularization strategies.

Provides numerical illustrations of convergence behaviors.

Abstract

In this work we investigate dynamical systems designed to approach the solution sets of inclusion problems involving the sum of two maximally monotone operators. Our aim is to design methods which guarantee strong convergence of trajectories towards the minimum norm solution of the underlying monotone inclusion problem. To that end, we investigate in detail the asymptotic behavior of dynamical systems perturbed by a Tikhonov regularization where either the maximally monotone operators themselves, or the vector field of the dynamical system is regularized. In both cases we prove strong convergence of the trajectories towards minimum norm solutions to an underlying monotone inclusion problem, and we illustrate numerically qualitative differences between these two complementary regularization strategies. The so-constructed dynamical systems are either of Krasnoselskii-Mann, of forward-backward type or of forward-backward-forward type, and with the help of injected regularization we demonstrate seminal results on the strong convergence of Hilbert space valued evolutions designed to solve monotone inclusion and equilibrium problems.