Trends to equilibrium for a nonlocal Fokker-Planck equation

F. Auricchio, G. Toscani, M. Zanella

Applied Math. Letters, 145, 108746, 2023. (Preprint arXiv)

We obtain equilibration rates for a one-dimensional nonlocal Fokker-Planck equation with time-dependent diffusion coefficient and drift, modeling the relaxation of a large swarm of robots, feeling each other in terms of their distance, towards the steady profile characterized by uniform spreading over a finite interval of the line. The result follows by combining entropy methods for quantifying the decay of the solution towards its quasi-stationary distribution, with the properties of the quasi-stationary profile.

On a kinetic description of Lotka-Volterra dynamics

G. Toscani, M. Zanella

Rivista di Matematica della UniversitĂ  di Parma, in press. (Preprint arXiv)

Owing to the analogies between the problem of wealth redistribution with taxation in a multi-agent society, we introduce and discuss a kinetic model describing the statistical distributions in time of the sizes of groups of biological systems with prey-predator dynamics.

While the evolution of the mean values is shown to be driven by a classical Lotka-Volterra system of differential equations, it is shown that the time evolution of the probability distributions of the size of groups of the two interacting species is heavily dependent both on a kinetic redistribution operator and the degree of randomness present in the system. Numerical experiments are given to clarify the time-behavior of the distributions of groups of the species.

Kinetic models for epidemic dynamics in the presence of opinion polarization

M. Zanella

Bulletin of Mathematical Biology, 85(5):36, 2023. (Preprint arXiv)

Understanding the impact of collective social phenomena in epidemic dynamics is a crucial task to effectively contain the disease spread. In this work, we build a mathematical description for assessing the interplay between opinion polarization and the evolution of a disease.

The proposed kinetic approach describes the evolution of aggregate quantities characterizing the agents belonging to epidemiologically relevant states and will show that the spread of the disease is closely related to consensus dynamics distribution in which opinion polarization may emerge. In the present modelling framework, microscopic consensus formation dynamics can be linked to macroscopic epidemic trends to trigger the collective adherence to protective measures. We conduct numerical investigations which confirm the ability of the model to describe different phenomena related to the spread of an epidemic.

A kinetic approach to consensus-based segmentation of biomedical images

R. F. Cabini, A. Pichiecchio, A. Lascialfari, S. Figini, M. Zanella

Preprint arXiv, 2022.

In this work, we apply a kinetic version of a bounded confidence consensus model to biomedical segmentation problems. In the presented approach, time-dependent information on the microscopic state of each particle/pixel includes its space position and a feature representing a static characteristic of the system, i.e. the gray level of each pixel. From the introduced microscopic model we derive a kinetic formulation of the model.

The large time behavior of the system is then computed with the aid of a surrogate Fokker-Planck approach that can be obtained in the quasi-invariant scaling. We exploit the computational efficiency of direct simulation Monte Carlo methods for the obtained Boltzmann-type description of the problem for parameter identification tasks. Based on a suitable loss function measuring the distance between the ground truth segmentation mask and the evaluated mask, we minimize the introduced segmentation metric for a relevant set of 2D gray-scale images. Applications to biomedical segmentation concentrate on different imaging research contexts.

On the optimal control of kinetic epidemic models with uncertain social features

J. Franceschi, A. Medaglia, M. Zanella.

Optim. Contr. Appl. Meth., 45(2): 494-522, 2024. (Preprint arXiv)

It is recognized that social heterogeneities in terms of the contact distribution have a strong influence on the spread of infectious diseases. Nevertheless, few data are available and their statistical description does not possess universal patterns and may vary spatially and temporally. It is therefore essential to design optimal control strategies, mimicking the effects of non-pharmaceutical interventions, to limit efficiently the number of infected cases.

In this work, starting from a recently introduced kinetic model for epidemiological dynamics that takes into account the impact of social contacts of individuals, we consider an uncertain contact formation dynamics leading to slim-tailed as well as fat-tailed distributions of contacts. Hence, we analyse the effects of an optimal control strategy of the system of agents. Thanks to classical methods of kinetic theory, we couple uncertainty quantification methods with the introduced mathematical model to assess the effects of social limitations. Finally, using the proposed modelling approach and starting from available data, we show the effectiveness of the proposed selective measures to dampen uncertainties together with the epidemic trends.

Stochastic Galerkin particle methods for kinetic equations of plasmas with uncertainties

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.

Micro-macro stochastic Galerkin methods for nonlinear Fokker-Plank equations with random inputs

G. Dimarco, L. Pareschi, M. Zanella.

Multiscale Model. Simul., 22(1): 527-560, 2024. (Preprint arXiv)

Nonlinear Fokker-Planck equations play a major role in modeling large systems of interacting particles with a proved effectiveness in describing real world phenomena ranging from classical fields such as fluids and plasma to social and biological dynamics.

Their mathematical formulation has often to face with physical forces having a significant random component or with particles living in a random environment which characterization may be deduced through experimental data and leading consequently to uncertainty-dependent equilibrium states. In this work, to address the problem of effectively solving stochastic Fokker-Planck systems, we will construct a new equilibrium preserving scheme through a micro-macro approach based on stochastic Galerkin methods. The resulting numerical method, contrarily to the direct application of a stochastic Galerkin projection in the parameter space of the unknowns of the underlying
Fokker-Planck model, leads to highly accurate description of the uncertainty dependent large time behavior. Several numerical tests in the context of collective behavior for social and life sciences are presented to assess the validity of the present methodology against standard ones.

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

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

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

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.