Strong convergence of an inertial Tikhonov regularized dynamical system governed by a maximally comonotone operator

TL;DR

This paper analyzes a dynamical system in a Hilbert space governed by a maximally comonotone operator, proving strong convergence to the minimal norm solution and establishing convergence rates under Tikhonov regularization.

Contribution

It introduces a new inertial Tikhonov regularized dynamical system with proven strong convergence and convergence rates for solutions of maximally comonotone operators.

Findings

Trajectory converges strongly to the minimum norm solution.

Convergence rates are established for specific regularization parameters.

Numerical example validates the theoretical results.

Abstract

In a Hilbert framework, we consider an inertial Tikhonov regularized dynamical system governed by a maximally comonotone operator, where the damping coefficient is proportional to the square root of the Tikhonov regularization parameter. Under an appropriate setting of the parameters, we prove the strong convergence of the trajectory of the proposed system towards the minimum norm element of zeros of the underlying maximally comonotone operator. When the Tikhonov regularization parameter reduces to $\frac{1}{t^q}$ with $0<q<1$, we further establish some convergence rate results of the trajectories. Finally, the validity of the proposed dynamical system is demonstrated by a numerical example.