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 ddotx(t) + dfracαtdotx(t) + nabla φ(x(t)) + β nabla2 φ(x(t))dotx(t)=0, where α and β are positive parameters, and φ:ℝn → ℝ is a smooth convex function. If φ 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 citeJMLR:v17:15-084, obtained in the strongly convex case for α=3 and β=0. Preliminary numerical experiments suggest that both adding a positive Hessian driven damping parameter β, and implementing the restart scheme help improve the performance of the dynamics and corresponding iterative algorithms as means to approximate minimizers of φ.