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

Advances in Nonlinear Analysis

article

Radu Ioan Boţ¹  Sorin-Mihai Grad¹  Dennis Meier¹  Mathias Staudigl² 

[1] Faculty of Mathematics, University of Vienna;Maastricht University, Department of Data Science and Knowledge Engineering

关键词: monotone inclusions;    dynamical systems;    Tikhonov regularization;    asymptotic analysis;   

DOI  :  10.1515/anona-2020-0143

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

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.

【 授权许可】

CC BY