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

2019/11/12 by Radu Ioan Boţ, Sorin‐Mihai Grad, Sorin-Mihai Grad +2 · 34 citations
Computer Science · Mathematics · #Applied mathematics #Computer science #Contact Mechanics and Variational Inequalities #Dynamical systems theory #Geometry #Hilbert space #Inverse problem #Mathematical analysis #Mathematics #Monotone polygon #Norm (philosophy) #Numerical methods in inverse problems #Optimization and Variational Analysis #Physics #Pure mathematics #Regularization (linguistics) #Strongly monotone #Tikhonov regularization #cs.NA #math.DS #math.FA #math.NA #math.OC #msc:34G25 #msc:37N40 #msc:47H05 #msc:90C25

paper · pdf · doi:10.1515/anona-2020-0143

published in Advances in Nonlinear Analysis 10(1), 450-476 (De Gruyter) · 30 pages, 21 figures

arxiv created 2019/11/12 · openalex created_date 2019/11/22 · openalex publication_date 2020/08/25 · arxiv updated 2020/08/31 · openalex updated_date 2026/08/06

Abstract

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 Krasnoselskiĭ-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.

Citations

• Splitting Algorithms for the Sum of Two Nonlinear Operators
• Tikhonov-like regularization of dynamical systems associated with nonexpansive operators defined in closed and convex sets
• A Dynamical Systems Perspective on Nesterov Acceleration
• On the convergence of gradient-like flows with noisy gradient input
• Variational inequalities