On the strong convergence of continuous Newton-like inertial dynamics with Tikhonov regularization for monotone inclusions
Radu Ioan Boţ, Ernö Robert Csetnek, Szilárd Csaba László
Abstract
In a Hilbert space H, we study the convergence properties of the trajectories of a Newton-like inertial dynamical system with a Tikhonov regularization term governed by a general maximally monotone operator A:H→2H. The maximally monotone operator enters the dynamics via its Yosida approximation with an appropriate adjustment of the Yosida regularization parameter, by adopting an approach introduced by Attouch-Peypouquet (Math. Prog., 2019) and further developed by Attouch-László (Set-Valued Var. Anal., 2021). We obtain fast rates of convergence for the velocity and the Yosida regularization term towards zero, while the generated trajectories converge weakly towards a zero of A or, depending on the system parameters, strongly towards the zero of minimum norm of A. Our analysis reveals that the damping coefficient, the Yosida regularization parameter and the Tikhonov parametrization are strongly correlated.