Strong convergence and fast rates for systems with Tikhonov regularization

Abstract

We introduce and investigate the asymptotic behaviour of the trajectories of a second order dynamical system with Tikhonov regularization for solving a monotone equation with single valued, monotone and continuous operator acting on a real Hilbert space. We consider a vanishing damping which is correlated with the Tikhonov parameter and which is in line with recent developments in the literature on this topic. A correction term which involves the time derivative of the operator along the trajectory is also involved in the system and makes the link with Newton and Levenberg-Marquardt type methods. We obtain strong convergence of the trajectory to the minimal norm solution and fast convergence rates for the velocity and a quantity involving the operator along the trajectory. The rates are very closed to the known fast convergence results for systems without Tikhonov regularization, the novelty with respect to this is that we also obtain strong convergence of the trajectory to the minimal norm solution. As an application we introduce a primal-dual dynamical system for solving linearly constrained convex optimization problems, where the strong convergence of the trajectories is highlighted together with fast convergence rates for the feasibility measure and function values.