Giacomo Albi, Lorenzo Pareschi, Mattia Zanella.
The study of formations and dynamics of opinions leading to the so called opinion consensus is one of the most important areas in mathematical modeling of social sciences. Following the Boltzmann type control recently introduced in [G. Albi, M. Herty, L. Pareschi arXiv:1401.7798], we consider a group of opinion leaders which modify their strategy accordingly to an objective functional with the aim to achieve opinion consensus. The main feature of the Boltzmann type control is that, thanks to an instantaneous binary control formulation, it permits to embed the minimization of the cost functional into the microscopic leaders interactions of the corresponding Boltzmann equation. The related Fokker-Planck asymptotic limits are also derived which allow to give explicit expressions of stationary solutions. The results demonstrate the validity of the Boltzmann type control approach and the capability of the leaders control to strategically lead the followers opinion.
Giacomo Albi, Michael Herty, Lorenzo Pareschi
In this paper an optimal control problem for a large system of interacting agents is considered using a kinetic perspective. As a prototype model we analyze a microscopic model of opinion formation under constraints. For this problem a Boltzmann-type equation based on a model predictive control formulation is introduced and discussed.
In particular, the receding horizon strategy permits to embed the minimization of suitable cost functional into binary particle interactions. The corresponding Fokker-Planck asymptotic limit is also derived and explicit expressions of stationary solutions are given. Several numerical results showing the robustness of the present approach are finally reported.
We consider the development of implicit-explicit time integration schemes for optimal control problems governed by the Goldstein-Taylor model. In the diffusive scaling this model is a hyperbolic approximation to the heat equation. We investigate the relation of time integration schemes and the formal Chapman-Enskog type limiting procedure.
For the class of stiffly accurate implicit-explicit Runge-Kutta methods (IMEX) the discrete optimality system also provides a stable numerical method for optimal control problems governed by the heat equation. Numerical examples illustrate the expected behavior.
G. Albi, D. Balagué, J. A. Carrillo, J. von Brecht (19/04/2013 arxiv.org/abs/1304.5459)
We study the linear stability of flock and mill ring solutions of two individual based models for biological swarming. The individuals interact via a nonlocal interaction potential that is repulsive in the short range and attractive in the long range. We relate the instability of the flock rings with the instability of the ring solution of the first order model. We observe that repulsive-attractive interactions lead to new configurations for the flock rings such as clustering and fattening formation. Finally, we numerically explore mill patterns arising from this kind of interactions together with the asymptotic speed of the system.
Related page with video & simulations.
Daniel Balague’s Homepage José A. Carillo’s Homepage James von Brecht’s Homepage
Giacomo Albi, Lorenzo Pareschi
(27 /09/2012 arXiv:1210.1172)
In nature self-organized systems as flock of birds, school of fishes or herd of sheeps have to deal with the presence of external agents such as predators or leaders which modify their internal dynamic. Such situations take into account a large number of individuals with their own social behavior which interact with a few number of other individuals acting as external point source forces. Starting from the microscopic description we derive the kinetic model through a mean-field limit and finally the macroscopic system through a suitable hydrodynamic limit.
Related page with videos & simulations.
Lorenzo Pareschi’s talk at BIRS, Banff Canada, april 2013.
Giacomo Albi, Lorenzo Pareschi
Microscopic models of flocking and swarming takes in account large numbers of interacting individuals. Numerical resolution of large flocks implies huge computational costs. Typically for $N$ interacting individuals we have a cost of $O(N^2)$. We tackle the problem numerically by considering approximated binary interaction dynamics described by kinetic equations and simulating such equations by suitable stochastic methods. This approach permits to compute approximate solutions as functions of a small scaling parameter at a reduced complexity of $O(N)$ operations. Several numerical results show the efficiency of the algorithms proposed.