Structure preserving schemes for nonlinear Fokker-Planck equations and applications

SW_pic Lorenzo Pareschi, Mattia Zanella                                        (Preprint arXiv)

In this paper we focus on the construction of numerical schemes for nonlinear Fokker-Planck equations that preserve the structural properties, like non negativity of the solution, entropy dissipation and large time behavior. The methods here developed are second order accurate, they do not require any restriction on the mesh size and are capable to capture the asymptotic steady states with arbitrary accuracy. These properties are essential for a correct description of the underlying physical problem. Applications of the schemes to several nonlinear Fokker-Planck equations with nonlocal terms describing emerging collective behavior in socio-economic and life sciences are presented

Recent advances in opinion modeling: control and social influence

Test3g_t5000Giacomo Albi, Lorenzo Pareschi, Giuseppe Toscani, Mattia Zanella (Preprint arXiv, to appear in

We survey some recent developments on the mathematical modeling of opinion dynamics. After an introduction on opinion modeling through interacting multi-agent systems described by partial differential equations of kinetic type, we focus our attention on two major advancements: optimal control of opinion formation and influence of additional social aspects, like conviction and number of connections in social networks, which modify the agents’ role in the opinion exchange process.

Kinetic models of collective decision-making in the presence of equality bias


Lorenzo Pareschi, Pierluigi Vellucci, Mattia Zanella (Physica A: Statistical Mechanics and its Applications, 10(1):1-32, 2017. Preprint arXiv)

We introduce and discuss kinetic models describing the influence of the competence in the evolution of decisions in a multi-agent system. The original exchange mechanism, which is based on the human tendency to compromise and change opinion through self-thinking, is here modified to include the role of the agents’ competence. In particular, we take into account the agents’ tendency to behave in the same way as if they were as good, or as bad, as their partner: the so-called equality bias. This occurred in a situation where a wide gap separated the competence of group members. We discuss the main properties of the kinetic models and numerically investigate some examples of collective decision under the influence of the equality bias. The results confirm that the equality bias leads the group to suboptimal decisions.

Opinion dynamics over complex networks: kinetic modelling and numerical methods


Giacomo Albi, Lorenzo Pareschi, Mattia Zanella (Kinetic and Related Models, 10(1): 1-32, 2017. Preprint arXiv)

In this paper we consider the modeling of opinion dynamics over time dependent large scale networks. A kinetic description of the agents’ distribution over the evolving network is considered which combines an opinion update based on binary interactions between agents with a dynamic creation and removal process of new connections. The number of connections of each agent influences the spreading of opinions in the network but also the way connections are created is influenced by the agents’ opinion. The evolution of the network of connections is studied by showing that its asymptotic behavior is consistent both with Poisson distributions and truncated power-laws. In order to study the large time behavior of the opinion dynamics a mean field description is derived which allows to compute exact stationary solutions in some simplified situations. Numerical methods which are capable to describe correctly the large time behavior of the system are also introduced and discussed. Finally, several numerical examples showing the influence of the agents’ number of connections in the opinion dynamics are reported.

Performance bounds for the mean-field limit of constrained dynamics

performance_boundplotMichael Herty, Mattia Zanella (Discrete and Continuous Dynamical Systems – Series A, 37(4): 2023-2043, 2017. Preprint arXiv)

In this work we are interested in the mean-field formulation of kinetic models under control actions where the control is formulated through a model predictive control strategy (MPC) with varying horizon. The relation between the (usually hard to compute) optimal control and the MPC approach is investigated theoretically in the mean-field limit. We establish a computable and provable bound on the difference in the cost functional for MPC controlled and optimal controlled system dynamics in the mean-field limit. The result of the present work extends previous findings for systems of ordinary differential equations. Numerical results in the mean-field setting are given.

On the optimal control of opinion dynamics on evolving networks


immagine_sitoGiacomo Albi, Lorenzo Pareschi, Mattia Zanella (To appear in IFIP TC7 2015 Proceedings. Preprint arXiv)

In this work we are interested in the modelling and control of opinion dynamics spreading on a time evolving network with scale-free asymptotic degree distribution. The mathematical model is formulated as a coupling of an opinion alignment system with a probabilistic description of the network. The optimal control problem aims at forcing consensus over the network, to this goal a control strategy based on the degree of connection of each agent has been designed. A numerical method based on a model predictive strategy is then developed and different numerical tests are reported. The results show that in this way it is possible to drive the overall opinion toward a desired state even if we control only a suitable fraction of the nodes.

Uncertainty Quantification in Control Problems for Flocking Models

Giacomo Albi, Lorenzo Pareschi, Mattia Zanella (Mathematical Problems in Engineering, Vol. 2015. Open Access. Preprint arXiv)

In this paper the optimal control of flocking models with random inputs is investigated from a numerical point of view. The effect of uncertainty in the interaction parameters is studied for a Cucker-Smale type model using a generalized polynomial chaos (gPC) approach. Numerical evidence of threshold effects in the alignment dynamic due to the random parameters is given. The use of a selective model predictive control permits to steer the system towards the desired state even in unstable regimes.

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Form and Urban Change. An urban morphology study of five gentrified neighbourhoods in London

baAlessandro Venerandi, Mattia Zanella, Ombretta Romice, Sergio Porta (To appear in Environment and Planning B)              

Many socioeconomic studies have been carried out to explain the phenomenon of gentrification. Although results of these works shed light on the process around this phenomenon, a perspective which focuses on the relationship between city form and gentrification is still missing. With this paper we try to address this gap by studying and comparing, through classic methods of mathematical statistics, morphological features of five London gentrified neighbourhoods. Outcomes confirm that areas which have undergone gentrification display similar and recognizable morphological patterns in terms of urban type and geographical location of main and local roads as well as businesses. These initial results confirm findings from previous research in urban sociology, and highlight the role of urban form in contributing to shape dynamics of non-spatial nature in cities.

Philip Ball’s comment on The Guardian: Logo-GuardianGentrification is a natural evolution | Philip Ball | Comment is free |




Stochastic modeling and simulation of ion transport through channels

Fig6d_grandeDaniela Morale, Mattia Zanella, Vincenzo Capasso, Willi Jäger                                                                                            (SIAM Journal on Multiscale Modeling & Simulation, vol. 14(1): 113-137. Preprint arXiv)

Ion channels are of major interest and form an area of intensive
research in the fields of biophysics and medicine since they control many vital physiological functions. The aim of this work is on one hand to propose a fully stochastic and discrete model describing the main characteristics of a multiple channel system. The movement of the ions is coupled, as usual, with a Poisson equation for the electrical field; we have considered, in addition, the influence of exclusion forces. On the other hand, we have discussed about the nondimensionalization of the stochastic system by using real physical parameters, all supported by numerical simulations. The specific features of both cases of micro- and nanochannels have been taken in due consideration with particular attention to the latter case in order to show that it is necessary to consider a discrete and stochastic model for ions movement inside the channels.

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Boltzmann-type control of opinion consensus through leaders

Kinetic_density_leaders_12 Giacomo Albi, Lorenzo Pareschi, Mattia Zanella    

(Phil. Trans. R. Soc. A 2014 372 20140138. Preprint arXiv)

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.


Article on the Italian blog Gli Stati Generali: Manuale per un leader: strategie matematiche di controllo dell’opinione pubblica.


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