In a Hilbert setting, our study focuses on the dynamical system introduced by Su-Boyd-Candès as a low resolution ODE of Nesterov’s accelerated gradient method (NAG). This inertial system, denoted by ${\mathrm{(AVD)}}_{\alpha }$ , is driven by the gradient of the function $f$ to be minimized, and is damped with an asymptotic vanishing coefficient of the form $\alpha /t$ , with $\alpha \geq 3$ . Taking $\alpha $ large enough plays a crucial role in the asymptotic convergence properties of the trajectories. For a general convex function $f$ , taking $\alpha >3$ guarantees the asymptotic convergence rate of the values $o \left ( 1/t^{2} \right )$ , as well as the convergence of the trajectories towards optimal solutions. For strongly convex $f$ , the asymptotic rate of convergence is of order $1/t^{\frac{2\alpha }{3}} $ , which increases with $\alpha $ . To analyze the effect of the parameter $\alpha $ in the convergence properties of ${\mathrm{(AVD)}}_{\alpha }$ , we show that a judicious time scaling of ${\mathrm{(AVD)}}_{\alpha }$ produces trajectories close to those of the continuous steepest descent method associated with $f$ when $\alpha $ is sufficiently large. This limiting process involves a singular perturbation property, as we move from a second-order evolution equation to a first-order one. This transition enables us to understand the change in the rate of convergence from $1/t$ to $1/t^{2}$ between the steepest descent method and (NAG). Based on a complexity analysis over a finite time interval, new results are obtained regarding the optimal tuning of the parameter $\alpha $ and the involved constants $C_{\alpha }$ in the estimations. Numerical experiments have been conducted to illustrate and confirm the theoretical results.
