Abstract : In recent years, kinetic-type models emerged to be a powerful mathematical framework for the description of emerging patterns in systems composed by a large number of agents. Furthermore, the natural multiscale nature of these equations, linking microscopic unobservable social forces to macroscopic measurable patterns, permits an efficient investigation of collective phenomena in a heterogeneity of disciplines, like biology, social sciences and robotics. In this minisymposium we collect novel perspectives from experts actively working on these research problems.
Abstract : We are interested in the construction of numerical methods for constrained high-dimensional constrained nonlinear optimization problems by gradient free techniques. Gradients are replaced by particle approximations and recently different methods have been proposed, e.g. consensus-based, swarm-based or ensemble Kalman based methods. We discuss recent extensions to the constrained and the parametric case as well as their corresponding mean field descriptions in the many particle limit. Those allow to show convergence as well as the analysis of properties of the new algorithm. Several numerical examples, also in high dimensions, illustrate the theoretical findings as well as the performance of those methods.
[03822] Parameter estimation for macroscopic pedestrian dynamics models using microscopic data
Format : Talk at Waseda University
Author(s) :
Susana Gomes (University of Warwick)
Abstract : I will present a framework for estimating relevant parameters for pedestrian dynamics by fitting a macroscopic model for crowd dynamics using data from pedestrian trajectories. The model couples a density dependent stochastic differential equation, to a nonlinear partial differential equation for the density via the fundamental diagram. I will discuss identifiability of the parameters, introduce optimisation and Bayesian methods to perform the identification, and analyse the performance of the proposed methodology in various realistic situations.
[03518] Navigation system based routing strategies in traffic flows on networks
Format : Talk at Waseda University
Author(s) :
Adriano Festa (Politecnico di Torino)
Abstract : Navigation choices play an important role in modeling and forecasting traffic flows on road networks. We introduce a macroscopic differential model coupling a conservation law with a Hamilton-Jacobi equation to respectively model the nonlinear transportation process and the strategic choices of users. Furthermore, the model is adapted to the multi-population case, where every population differs in the level of traffic information about the system.
[02357] Uncertainty quantification in vehicular traffic models
Format : Talk at Waseda University
Author(s) :
Elisa Iacomini (University of Ferrara, Department of Mathematics and Computer Science)
Abstract : Traffic models have been widely studied, however limitations for obtaining reliable forecasts are still present. Recently it has been pointed out how traffic is exposed to the presence of uncertainties. In this talk, starting from the hierarchy between microscopic, kinetic and macroscopic scales, we will investigate the propagation of uncertainties through the models. Connections between the scales will be presented in the stochastic scenario and numerical simulations will be performed.
[04654] On a kinetic Elo rating model for players with dynamical strength
Format : Talk at Waseda University
Author(s) :
Bertram Düring (University of Warwick)
Abstract : We discuss a new kinetic rating model for a large number of players, which is motivated by the well-known Elo
rating system. Each player is characterised by an intrinsic strength and a rating, which are both updated
after each game. We state and analyse the respective Boltzmann-type equation and derive the corresponding
nonlinear, nonlocal Fokker-Planck equation. We investigate the existence of solutions to the Fokker-Planck
equation and discuss their behaviour in the long time limit. Furthermore, we illustrate the dynamics of the
Boltzmann and Fokker-Planck equation with various numerical experiments.
[04235] Nonlocal approximation of nonlinear diffusion equations and cross-diffusion systems
Format : Talk at Waseda University
Author(s) :
Antonio Esposito (University of Oxford)
Martin Burger (University of Hamburg)
José Antonio Carrillo (University of Oxford)
Jeremy S.-H. Wu (UCLA)
Abstract : In this talk I will discuss the connection between a class of nonlocal PDEs and nonlinear diffusion equations, including porous medium PDEs and cross-diffusion systems. As byproduct of this link, one can obtain a rigorous deterministic particle approximation for the PDEs considered. The analysis is based on a suitable regularisation of the associated free energy using gradient flow techniques. However, the strategy proposed relies on a discretisation scheme, so-called JKO, which can be slightly modified in order to extend the results to PDEs without gradient flow structure. In particular, it does not require convexity of the associated energies. The talk is based on two joint works with M. Burger (FAU Erlangen-Nuremberg), and J. A. Carrillo (Oxford) and J. Wu (UCLA).
[04374] Kinetic models for multi-agent systems with multiple microscopic states
Format : Talk at Waseda University
Author(s) :
Nadia Loy (Politecnico di Torino)
Abstract : In this talk we present a class of kinetic models describing interactions among individuals having multiple microscopic states. We shall consider microscopic states evolving according to both stochastic dependent and independent processes. In particular, we shall consider interacting agents who are divided into multiple sub-populations. As such, the agents are not indistinguishable, as classically assumed in kinetic theory, within the whole population.
A general framework allowing to describe binary interactions and transfers among different sub-groups by deriving the model from microscopic stochastic processes will be presented. We shall discuss formal results concerning existence, uniqueness and equilibria. Moreover, we shall illustrate applications to wealth exchange models with migration.
[03362] Modelling coevolutionary dynamics in heterogeneous SI epidemiological systems across scales
Format : Talk at Waseda University
Author(s) :
Elisa Paparelli (Politecnico di Torino)
Tommaso Lorenzi (Politecnico di Torino)
Andrea Tosin (Politecnico di Torino)
Abstract : We present a new structured compartmental epidemiological model for the coevolutionary dynamics between susceptible and infectious individuals. Specifically, continuous structuring variables capture interindividual variability in resistance to infection and viral load. The model comprises a system of integro-differential equations providing a Boltzmann-type kinetic description of corresponding stochastic particle dynamics. We discuss a formal derivation of this model from the underlying particle dynamics and present analytical and numerical results on the long-time behaviour of its solutions.
[05559] The Collisional Particle-In-Cell Method for the Vlasov-Maxwell-Landau System
Format : Talk at Waseda University
Author(s) :
Rafael Bailo (University of Oxford, Mathematical Institute)
José Antonio Carrillo (University of Oxford, Mathematical Institute)
Jingwei Hu (University of Washington)
Abstract : In this talk we will present an extension of the classical Particle-In-Cell (PIC) method
for plasmas which can account for the collisional effects modelled by the Landau operator.
The method is derived form the gradient-flow formulation of the Landau equation, thereby
preserving the collision invariants and the entropy structure. We will discuss the derivation
and implementation of the method, as well as several numerical examples to showcase the
effects of collisions in plasma simulations.
