A speed restart scheme for a dynamics with Hessian driven damping

In this paper, we analyze a speed restarting scheme for the dynamical system given by $$\ddot{x}(t) + \dfrac{\alpha}{t}\dot{x}(t) + \nabla \phi(x(t)) + \beta \nabla^2 \phi(x(t))\dot{x}(t)=0,$$ where $\alpha$ and $\beta$ are positive parameters, and $\phi:\mathbb{R}^n \to \mathbb{R}$ is a smooth convex function. If $\phi$ has quadratic growth, we establish a linear convergence rate for the function values along the restarted trajectories. As a byproduct, we improve the results obtained by Su, Boyd and Cand\`es \cite{JMLR:v17:15-084}, obtained in the strongly convex case for $\alpha=3$ and $\beta=0$. Preliminary numerical experiments suggest that both adding a positive Hessian driven damping parameter $\beta$, and implementing the restart scheme help improve the performance of the dynamics and corresponding iterative algorithms as means to approximate minimizers of $\phi$.