Fast optimization and algorithms for the charged particle dynamics with variable density

In this paper, we propose a new autonomous dissipative inertial continuous dynamics with variable density in the magnetic filed related to convex optimization problems. This system extends the class of Heavy Ball model with damping by featuring a time-dependent parameter inside the acceleration as the variable density and adding the magnetic coupled term [Formula: see text]. To accelerate the development of efficient optimization methods, our first significant contribution involves demonstrating the global existence and uniqueness of smooth solution for the system under consideration, given specific regularity conditions, by employing the Banach fixed point theorem. Additionally, we have proven the existence of a global attractor through a comprehensive convergence analysis. Our second pivotal contribution lies in establishing the convergence rate of corresponding algorithms that incorporate inertial characteristics. This is achieved by analyzing the discrete time versions of inertial dynamics within the magnetic field framework. Furthermore, we have elucidated the connection between these algorithms and the Heavy Ball model. The corresponding numerical experiments are conducted for the illustration of our new algorithms. Our results expand the ideas previously laid out in [W. P. Yan, Y. Tang and G. L. Yuan, Fast optimization of charged particle dynamics with damping, SIAM J. Optim. 34 (2024) 2287–2313].