This paper presents an analysis of an implicit Newton-type inertial method. The aim of this method is to find \(x\in \mathcal{H}\) such that \( \nabla f(x) + B(x)=0 \), where \(\mathcal{H}\) is a real Hilbert space, \(f\) is a continuously differentiable convex function and \(B\) a monotone and cocoercive operator.
The authors consider a dynamical system whose stationary points are solutions to the following problem: \[ \ddot{x}(t) + \gamma \dot{x}(t) +\partial f(x(t) +\beta_f \dot{x}(t)) +B(x(t)+\beta_b \dot{x}(t)) = 0. \] The terms \(\partial f(x(t) +\beta_f \dot{x}(t))\) and \(B(x(t)+\beta_b \dot{x}(t))\) correspond to first-order Taylor expansions, which relates this scheme to a Newtown-type scheme.
The authors show that the considered dynamical system is well behaved, and that, under reasonable assumptions, the system has a unique strong global solution for any given initial condition (Cauchy data) \((x(0),\dot{x}(0))\). Furthermore, they show that every solution trajectory of the dynamical system asymptotically converges to a solution of the original problem \( \nabla f(x) + B(x)=0 \).
In the second part of the paper, a splitting proximal algorithm is derived, from the discretisation of the system and it is shown that under some conditions the algorithm produces a sequence converging (weakly) to a solution. Several variations of this algorithm are considered, such as perturbed systems or finite difference approaches.
The article is concluded with a numerical section showing on the basis of a simple example how the proposed scheme (called iDINAM) leads to attenuated oscillations in the convergence of the solution, as compared to the case when \(\beta_f=\beta_b=0\).