Reduced variance random batch methods for nonlocal PDEs

L. Pareschi, M. Zanella

Preprint arXiv, 2023

Random Batch Methods (RBM) for mean-field interacting particle systems enable the reduction of the quadratic computational cost associated with particle interactions to a near-linear cost. The essence of these algorithms lies in the random partitioning of the particle ensemble into smaller batches at each time step.

The interaction of each particle within these batches is then evolved until the subsequent time step. This approach effectively decreases the computational cost by an order of magnitude while increasing the amount of fluctuations due to the random partitioning. In this work, we propose a variance reduction technique for RBM applied to nonlocal PDEs of Fokker-Planck type based on a control variate strategy. The core idea is to construct a surrogate model that can be computed on the full set of particles at a linear cost while maintaining enough correlations with the original particle dynamics. Examples from models of collective behavior in opinion spreading and swarming dynamics demonstrate the great potential of the present approach.

Uncertainty Quantification for the Homogeneous Landau-Fokker-Planck Equation via Deterministic Particle Galerkin methods

R. Bailo, J. A. Carrillo, A. Medaglia, M. Zanella

Preprint arXiv, 2023.

We design a deterministic particle method for the solution of the spatially homogeneous Landau equation with uncertainty. The deterministic particle approximation is based on the reformulation of the Landau equation as a formal gradient flow on the set of probability measures, whereas the propagation of uncertain quantities is computed by means of a sG representation of each particle.

This approach guarantees spectral accuracy in uncertainty space while preserving the fundamental structural properties of the model: the positivity of the solution, the conservation of invariant quantities, and the entropy production. We provide a regularity results for the particle method in the random space. We perform the numerical validation of the particle method in a wealth of test cases.

Effects of heterogeneous opinion interactions in many-agent systems for epidemic dynamics

S. Bonandin, M. Zanella

Networks and Heterogeneous Media, in press. (Preprint arXiv)

In this work we define a kinetic model for understanding the impact of heterogeneous opinion formation dynamics on epidemics. The considered many-agent system is characterized by nonsymmetric interactions which define a coupled system of kinetic equations for the evolution of the opinion density in each compartment.

In the quasi-invariant limit we may show positivity and uniqueness of the solution of the problem together with its convergence towards an equilibrium distribution exhibiting bimodal shape. The tendency of the system towards opinion clusters is further analyzed by means of numerical methods, which confirm the consistency of the kinetic model with its moment system whose evolution is approximated in several regimes of parameters.

Kinetic compartmental models driven by opinion dynamics: vaccine hesitancy and social influence

A. Bondesan, G. Toscani, M. Zanella

Mathematical Model and Methods in Applied Sciences, in press. (Preprint arXiv)

We propose a kinetic model for understanding the link between opinion formation phenomena and epidemic dynamics. The recent pandemic has brought to light that vaccine hesitancy can present different phases and temporal and spatial variations, presumably due to the different social features of individuals.

The emergence of patterns in societal reactions permits to design and predict the trends of a pandemic. This suggests that the problem of vaccine hesitancy can be described in mathematical terms, by suitably coupling a kinetic compartmental model for the spreading of an infectious disease with the evolution of the personal opinion of individuals, in the presence of leaders. The resulting model makes it possible to predict the collective compliance with vaccination campaigns as the pandemic evolves and to highlight the best strategy to set up for maximizing the vaccination coverage. We conduct numerical investigations which confirm the ability of the model to describe different phenomena related to the spread of an epidemic.

Impact of interaction forces in first order many-agent systems for swarm manufacturing

F. Auricchio, M. Carraturo, G. Toscani, M. Zanella.

Disc. Cont. Dyn. Syst. – S, 17(1):78-97, 2024. (Preprint arXiv)

We study the large time behavior of a system of interacting agents modeling the relaxation of a large swarm of robots, whose task is to uniformly cover a portion of the domain by communicating with each other in terms of their distance.

To this end, we generalize a related result for a Fokker-Planck-type model with a nonlocal discontinuous drift and constant diffusion, recently introduced by three of the authors, of which the steady distribution is explicitly computable. For this new nonlocal Fokker-Planck equation, existence, uniqueness and positivity of a global solution are proven, together with precise equilibration rates of the solution towards its quasi-stationary distribution. Numerical experiments are designed to verify the theoretical findings and explore possible extensions to more complex scenarios.

Particle simulation methods for the Landau-Fokker-Planck equation with uncertain data

A. Medaglia, L. Pareschi, M. Zanella

Journal of Computational Physics, in press. (Preprint arXiv)

The design of particle simulation methods for collisional plasma physics has always represented a challenge due to the unbounded total collisional cross section, which prevents a natural extension of the classical Direct Simulation Monte Carlo (DSMC) method devised for the Boltzmann equation.

One way to overcome this problem is to consider the design of Monte Carlo algorithms that are robust in the so-called grazing collision limit. In the first part of this manuscript, we will focus on the construction of collision algorithms for the Landau-Fokker-Planck equation based on the grazing collision asymptotics and which avoids the use of iterative solvers. Subsequently, we discuss problems involving uncertainties and show how to develop a stochastic Galerkin projection of the particle dynamics which permits to recover spectral accuracy for smooth solutions in the random space. Several classical numerical tests are reported to validate the present approach.

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 Università 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.