Structure preserving schemes for the continuum Kuramoto model: phase transitions

José A. Carrillo, Young-Pil Choi, Lorenzo Pareschi  (13/3/2018 preprint arXiv:1803.03886)

The construction of numerical schemes for the Kuramoto model is challenging due to the structural properties of the system which are essential in order to capture the correct physical behavior, like the description of stationary states and phase transitions. Additional difficulties are represented by the high dimensionality of the problem in presence of multiple frequencies.

Kinetic models for optimal control of wealth inequalities

Bertram Düring, Lorenzo Pareschi, Giuseppe Toscani (6/3/2018, preprint arXiv:1803.02171)

We introduce and discuss optimal control strategies for kinetic models for wealth distribution in a simple market economy, acting to minimize the variance of the wealth density among the population. Our analysis is based on a model predictive control approximation of the microscopic agents' dynamic and results in an alternative theoretical approach to the taxation and redistribution policy.

Boltzmann games in heterogeneous consensus dynamics

Giacomo Albi, Lorenzo Pareschi, Mattia Zanella (12/12/2017 preprint arXiv:1712.03224)

We consider a constrained hierarchical opinion dynamics in the case of leaders' competition and with complete information among leaders. Each leaders' group tries to drive the followers' opinion towards a desired state accordingly to a specific strategy. By using the Boltzmann-type control approach we analyze the best-reply strategy for each leaders' population. Derivation of the corresponding Fokker-Planck model permits to investigate the asymptotic behaviour of the solution.

Particle based gPC methods for mean-field models of swarming with uncertainty

José Antonio Carrillo, Lorenzo Pareschi, Mattia Zanella (5/12/2017 preprint arXiv:1712.01677) to appear in Comm. Comp. Phys.

In this work we focus on the construction of numerical schemes for the approximation of stochastic mean--field equations which preserve the nonnegativity of the solution. The method here developed makes use of a mean-field Monte Carlo method in the physical variables combined with a generalized Polynomial Chaos (gPC) expansion in the random space. In contrast to a direct application of stochastic-Galerkin methods, which are highly accurate but lead to the loss of positivity, the