Fast Convergence of Inertial Gradient Dynamics with Multiscale Aspects

In this paper, the asymptotic properties as $$ t\rightarrow +\infty $$ t → + ∞ of the following second-order differential equation in a Hilbert space $$ {\mathcal {H}} $$ H are studied, $$\begin{aligned} \ddot{x}(t)+\gamma (t){\dot{x}}(t)+\beta (t)\Big (\nabla \Phi (x(t))+\epsilon (t)\nabla U(x(t))\Big )=0, \end{aligned}$$ x ¨ ( t ) + γ ( t ) x ˙ ( t ) + β ( t ) ( ∇ Φ ( x ( t ) ) + ϵ ( t ) ∇ U ( x ( t ) ) ) = 0 , where $$ \Phi ,U:{\mathcal {H}}\rightarrow {\mathbb {R}} $$ Φ , U : H → R are convex differentiable, $$ \gamma (t) $$ γ ( t ) is a positive damping coefficient, $$ \beta (t) $$ β ( t ) is a time scale coefficient and $$ \epsilon (t) $$ ϵ ( t ) is a positive nonincreasing function, $$\gamma (t)$$ γ ( t ) , $$ \beta (t) $$ β ( t ) and $$ \epsilon (t) $$ ϵ ( t ) are all continuously differentiable. This system has applications in the fields of mechanics and optimization. Based on the proper tuning of $$ \gamma (t) $$ γ ( t ) and $$ \beta (t) $$ β ( t ) , we obtain the convergence rates for the values, and the conclusion is that, under the different conditions, the trajectories either converge to one minimizer of $$ \Phi $$ Φ weakly, or converge to one common minimizer of $$ \Phi $$ Φ and U weakly. When $$ \epsilon (t) $$ ϵ ( t ) tends to 0 as t goes to infinity, under the condition that $$ \Phi $$ Φ or U is convex, the trajectories converge to the unique minimizer of $$ \Phi $$ Φ , or the unique minimizer of U, respectively. Finally, some particular cases are examined, and some numerical experiments are conducted to illustrate our main results.