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The study of collective behavior in multi-agent systems has attracted the attention of many researchers due to its wide range of applications. Among them, the Cucker-Smale model was developed to study the phenomenon of flocking, and various types of extended models have been actively proposed and studied in recent decades. In this study, we address open questions of the Cucker--Smale model with norm-type Rayleigh friction: {\bf (i)} The positivity of the communication weight, {\bf (ii)} The convergence of the norm of the velocities of agents, {\bf (iii)} The direction of the velocities of agents. For problems (i) and (ii), we present the nonlinear Cucker--Smale model with norm-type Rayleigh friction, where the nonlinear Cucker--Smale model is generalized to a nonlinear model by applying a discrete $p$-Laplacian operator. For this model, we present conditions that guarantee that the norm for velocities of agents converges to 0 or a positive value, and we also show that the regular communication weight satisfies the conditions given in this study. In particular, we present a condition for the initial configuration to obtain that the norm of agent velocities converges to only some positive value. By contrast, problem (iii) is not solved by the norm-type nonlinear model. Thus, we propose a nonlinear Cucker--Smale model with a vector-type Rayleigh friction for problem (iii). In parallel to the first model, we show that the direction of the agents' velocities can be controlled by parameters in the nonlinear Cucker--Smale model with the vector-type Rayleigh friction., Comment: 26 pages

We introduce a fast solver for the phase field crystal (PFC) and functionalized Cahn-Hilliard (FCH) equations with periodic boundary conditions on a rectangular domain that features the preconditioned Nesterov accelerated gradient descent (PAGD) method. We discretize these problems with a Fourier collocation method in space, and employ various second-order schemes in time. We observe a significant speedup with this solver when compared to the preconditioned gradient descent (PGD) method. With the PAGD solver, fully implicit, second-order-in-time schemes are not only feasible to solve the PFC and FCH equations, but also do so more efficiently than some semi-implicit schemes in some cases where accuracy issues are taken into account. Benchmark computations of five different schemes for the PFC and FCH equations are conducted and the results indicate that, for the FCH experiments, the fully implicit schemes (midpoint rule and BDF2 equipped with the PAGD as a nonlinear time marching solver) perform better than their IMEX versions in terms of computational cost needed to achieve a certain precision. For the PFC, the results are not as conclusive as in the FCH experiments, which, we believe, is due to the fact that the nonlinearity in the PFC is milder nature compared to the FCH equation. We also discuss some practical matters in applying the PAGD. We introduce an averaged Newton preconditioner and a sweeping-friction strategy as heuristic ways to choose good preconditioner parameters. The sweeping-friction strategy exhibits almost as good a performance as the case of the best manually tuned parameters.

We develop a theoretical foundation for the application of Nesterov's accelerated gradient descent method (AGD) to the approximation of solutions of a wide class of partial differential equations (PDEs). This is achieved by proving the existence of an invariant set and exponential convergence rates when its preconditioned version (PAGD) is applied to minimize locally Lipschitz smooth, strongly convex objective functionals. We introduce a second-order ordinary differential equation (ODE) with a preconditioner built-in and show that PAGD is an explicit time-discretization of this ODE, which requires a natural time step restriction for energy stability. At the continuous time level, we show an exponential convergence of the ODE solution to its steady state using a simple energy argument. At the discrete level, assuming the aforementioned step size restriction, the existence of an invariant set is proved and a matching exponential rate of convergence of the PAGD scheme is derived by mimicking the energy argument and the convergence at the continuous level. Applications of the PAGD method to numerical PDEs are demonstrated with certain nonlinear elliptic PDEs using pseudo-spectral methods for spatial discretization, and several numerical experiments are conducted. The results confirm the global geometric and mesh size-independent convergence of the PAGD method, with an accelerated rate that is improved over the preconditioned gradient descent (PGD) method.