Category Archives: Kinetic modelling

Stochastic Galerkin particle methods for kinetic equations of plasmas with uncertainties

by Mattia Zanella

A. Medaglia, L. Pareschi, M. Zanella

Journal of Computational Physics, 479:112011, 2023. (Preprint arXiv)

The study of uncertainty propagation is of fundamental importance in plasma physics simulations. To this end, in the present work we propose a novel stochastic Galerkin (sG) particle methods for collisional kinetic models of plasmas under the effect of uncertainties.

This class of methods is based on a generalized polynomial chaos (gPC) expansion of the particles’ position and velocity. In details, we introduce a stochastic particle approximation for the Vlasov-Poisson system with a BGK term describing plasma collisions. A careful reformulation of such dynamics is needed to perform the sG projection and to obtain the corresponding system for the gPC coefficients. We show that the sG particle method preserves the main physical properties of the problem, such as conservations and positivity of the solution, while achieving spectral accuracy for smooth solutions in the random space. Furthermore, in the fluid limit the sG particle solver is designed to possess the asymptotic-preserving property necessary to obtain a sG particle scheme for the limiting Euler-Poisson system, thus avoiding the loss of hyperbolicity typical of conventional sG methods based on finite differences or finite volumes. We tested the schemes considering the classical Landau damping problem in the presence of both small and large initial uncertain perturbations, the two stream instability and the Sod shock tube problems under uncertainties. The results show that the proposed method is able to capture the correct behavior of the system in all test cases, even when the relaxation time scale is very small.

Fokker-Planck modeling of many-agent systems in swarm manufacturing: asymptotic analysis and numerical results

by Mattia Zanella

F. Auricchio, G. Toscani, M. Zanella

Communications in Mathematical Sciences, 21(6):1655-1677, 2023. (Preprint arXiv)

In this paper we study a novel Fokker-Planck-type model that is designed to mimic manufacturing processes through the dynamics characterizing a large set of agents. In particular, we describe a many-agent system interacting with a target domain in such a way that each agent/particle is attracted by the center of mass of the target domain with the aim to uniformly cover this zone.

To this end, we first introduce a mean-field model with discontinuous flux whose large time behavior is such that the steady state is globally continuous and uniform over a connected portion of the domain. We prove that a diffusion coefficient that guarantees that a given portion of mass enters in the target domain exists and that it is unique. Furthermore, convergence to equilibrium in 1D is provided through a reformulation of the initial problem involving a nonconstant diffusion function. The extension to 2D is explored numerically by means of recently introduced structure preserving methods for Fokker-Planck equations.

From agent-based models to the macroscopic description of fake-news spread: the role of competence in data-driven applications

by Mattia Zanella

J. Franceschi, L. Pareschi, M. Zanella

Partial Differential Equations and Applications, 3, 68, 2022. (Preprint arXiv)

Fake news spreading, with the aim of manipulating individuals’ perceptions of facts, is now recognized as a major problem in many democratic societies. Yet, to date, little has been understood about how fake news spreads on social networks, what the influence of the education level of individuals is, when fake news is effective in influencing public opinion, and what interventions might be successful in mitigating their effect.

In this paper, starting from the recently introduced kinetic multi-agent model with competence by the first two authors, we propose to derive reduced- order models through the notion of social closure in the mean-field approximation that has its roots in the classical hydrodynamic closure of kinetic theory. This approach allows to obtain simplified models in which the competence and learning of the agents maintain their role in the dynamics and, at the same time, the structure of such models is more suitable to be interfaced with data-driven applications. Examples of different Twitter-based test cases are described and discussed.

Monte Carlo stochastic Galerkin methods for non-Maxwellian kinetic models of multiagent systems with uncertainties

by Mattia Zanella

A. Medaglia, A. Tosin, M. Zanella

Partial Differential Equations and Applications, 3, 51, 2022. (Preprint arXiv)

In this paper, we focus on the construction of a hybrid scheme for the approximation of non- Maxwellian kinetic models with uncertainties. In the context of multiagent systems, the introduction of a kernel at the kinetic level is useful to avoid unphysical interactions.

The methods here proposed, combine a direct simulation Monte Carlo (DSMC) in the phase space together with stochastic Galerkin (sG) methods in the random space. The developed schemes preserve the main physical properties of the solution together with accuracy in the random space. The consistency of the methods is tested with respect to surrogate Fokker-Planck models that can be obtained in the quasi-invariant regime of parameters. Several applications of the schemes to non-Maxwellian models of multiagent systems are reported.

Effects of vaccination efficacy on wealth distribution in kinetic epidemic models

by Mattia Zanella

E. Bernardi, L. Pareschi, G. Toscani, M. Zanella

Entropy, 22:216, 2022. (Preprint arXiv)

The spreading of Covid-19 pandemic has highlighted the close link between economics and health in the context of emergency management. A widespread vaccination campaign is considered the main tool to contain the economic consequences. This paper will focus, at the level of wealth distribution modelling, on the economic improvements induced by the vaccination campaign in terms of its effectiveness rate. The economic trend during the pandemic is evaluated resorting to a mathematical model joining a classical compartmental model including vaccinated individuals with a kinetic model of wealth distribution based on binary wealth exchanges. The interplay between wealth exchanges and the progress of the infectious disease is realized by assuming on the one hand that individuals in different compartments act differently in the economic process and on the other hand that the epidemic affects risk in economic transactions. Using the mathematical tools of kinetic theory, it is possible to identify the equilibrium states of the system and the formation of inequalities due to the pandemic in the wealth distribution of the population. Numerical experiments highlight the importance of the vaccination campaign and its positive effects in reducing economic inequalities in the multi-agent society

Uncertainty quantification and control of kinetic models for tumour growth under clinical uncertainties

by Mattia Zanella

A. Medaglia, G. Colelli, L. Farina, A. Bacila, P. Bini, E. Marchioni, S. Figini, A. Pichiecchio, M. Zanella

International Journal of Non-Linear Mechanics, 141:103933. (Preprint arXiv)

In this work, we develop a kinetic model for tumour growth taking into account the effects of clinical uncertainties characterising the tumours’ progression.

The action of therapeutic protocols trying to steer the tumours’ volume towards a target size is then investigated by means of suitable selective-type controls acting at the level of cellular dynamics. By means of classical tools of statistical mechanics for many-agent systems, we are able to prove that it is possible to dampen clinical uncertainties across the scales. To take into account the scarcity of clinical data and the possible source of error in the image segmentation of tumours’ evolution, we estimated empirical distributions of relevant parameters that are considered to calibrate the resulting model obtained from real cases of primary glioblastoma. Suitable numerical methods for uncertainty quantification of the resulting kinetic equations are discussed and, in the last part of the paper, we compare the effectiveness of the introduced control approaches in reducing the variability in tumours’ size due to the presence of uncertain quantities.

A multi-agent description of the influence of higher education on social stratification

by Mattia Zanella

G. Dimarco, G. Toscani, M. Zanella

Journal of Economic Interaction and Coordination, in press. (Preprint arXiv)

We introduce and discuss a system of one-dimensional kinetic equations describing the influence of higher education in the social stratification of a multi-agent society.

The system is obtained by coupling a model for knowledge formation with a kinetic description of the social climbing in which the parameters characterizing the elementary interactions leading to the formation of a social elite are assumed to depend on the degree of knowledge/education of the agents. In addition, we discuss the case in which the education level of an individual is function of the position occupied in the social ranking. With this last assumption we obtain a fully coupled model in which knowledge and social status influence each other. In the last part, we provide several numerical experiments highlighting the role of education in reducing social inequalities and in promoting social mobility.

Kinetic modelling of epidemic dynamics: social contacts, control with uncertain data, and multiscale spatial dynamics

by Mattia Zanella

G. Albi, G. Bertaglia, W. Boscheri, G. Dimarco, L. Pareschi, G. Toscani, M. Zanella.

Predicting Pandemics in a Globally Connected World Vol.1, 2022. (Preprint arXiv)

In this survey we report some recent results in the mathematical modeling of epidemic phenomena through the use of kinetic equations.

We initially consider models of interaction between agents in which social characteristics play a key role in the spread of an epidemic, such as the age of individuals, the number of social contacts, and their economic wealth. Subsequently, for such models, we discuss the possibility of containing the epidemic through an appropriate optimal control formulation based on the policy maker’s perception of the progress of the epidemic. The role of uncertainty in the data is also discussed and addressed. Finally, the kinetic modeling is extended to spatially dependent settings using multiscale transport models that can characterize the impact of movement dynamics on epidemic advancement on both one-dimensional networks and realistic two-dimensional geographic settings.

Kinetic derivation of Aw-Rascle-Zhang-type traffic models with driver-assist vehicles

by Mattia Zanella

G. Dimarco, A. Tosin, M. Zanella

Journal of Statistical Physics, 186, 17, 2022. (Preprint arXiv)

In this paper, we derive second order hydrodynamic traffic models from kinetic-controlled equations for driver-assist vehicles. At the vehicle level we take into account two main control strategies synthesising the action of adaptive cruise controls and cooperative adaptive cruise controls. The resulting macroscopic dynamics fulfil the anisotropy condition introduced in the celebrated Aw-Rascle-Zhang model. Unlike other models based on heuristic arguments, our approach unveils the main physical aspects behind frequently used hydrodynamic traffic models and justifies the structure of the resulting macroscopic equations incorporating driver-assist vehicles. Numerical insights show that the presence of driver-assist vehicles produces an aggregate homogenisation of the mean flow speed, hich may also be steered towards a suitable desired speed in such a way that optimal flows and traffic stabilisation are reached

 

Kinetic-controlled hydrodynamics for multilane traffic models

by Mattia Zanella

R. Borsche, A. Klar, M. Zanella

Physica A: Statistical Mechanics and its Applications, 587: 126486, 2022. (Preprint arXiv)

We study the application of a recently introduced hierarchical description of traffic flow control by driver-assist vehicles to include lane changing dynamics. Lane-dependent feedback control strategies are implemented at the level of vehicles and the aggregate trends are studied by means of Boltzmann-type equations determining three different hydrodynamics based on the lane switching frequency. System of first order macroscopic equations describing the evolution of densities along the lanes are then consistently determined through a suitable closure strategy. Numerical examples are then presented to illustrate the features of the proposed hierarchical approach.