Accelerated Convergence of a Second-Order Dynamical System and its Application to Splitting Algorithms for Comonotone Inclusions

Abstract

This paper introduces a novel second-order dynamical system driven by a forward-backward splitting operator for solving the structured inclusion $0\in(A+B)(x)$ in a real Hilbert space, where $A$ is a maximal $ρ$-comonotone operator and $B$ is a $ν$-cocoercive operator. The well-posedness of the system is established, and Lyapunov analysis yields accelerated convergence rates of order $o(\frac{1}{t})$ for the velocity and $o(\frac{1}{t^2})$ for the forward-backward residual, together with weak convergence of the trajectories to $\operatorname{zer}(A+B)$. Temporal discretization further leads to a class of double inertial Halpern forward-backward splitting algorithms that encompasses the classical forward-backward splitting method and its inertial variants as special cases. Numerical experiments on split feasibility, sparse signal recovery, and image deblurring illustrate the effectiveness of the proposed algorithm.