A fast spectral particle method for the Landau equation

 

Giacomo Borghi, Lorenzo Pareschi (preprint arXiv:2609.23086)

We propose a fast deterministic particle method for the spatially homogeneous Landau equation. The method combines a particle representation of the solution with a Fourier approximation of the nonlinear collision flux. Nonuniform fast Fourier transforms are used to reconstruct the density from the particles and to evaluate the flux and density at the particle locations, while the convolutional structure of the flux enables its efficient computation by FFTs. For a fixed transform tolerance, the cost per time step is O(N+M^d logM), where N is the number of particles and M the number of Fourier modes per velocity dimension. We establish a consistency estimate for the reconstructed velocity field on regions where the reference density is bounded away from zero. The estimate separates the spectral truncation error from the particle-density reconstruction error, showing how the latter can dominate for sufficiently smooth densities.

Control variates with neural surrogates for uncertainty quantification in kinetic equations

Wei Chen, Giacomo Dimarco, Lorenzo Pareschi (preprint arXiv:2608.01360)

Efficient uncertainty quantification for kinetic equations with random inputs is challenging because it requires repeated simulations of high-dimensional models, such as the Boltzmann, Landau, and related collisional equations, whose computational cost can quickly become prohibitive. Multifidelity control variates address this difficulty by coupling a small number of high-fidelity simulations with many evaluations of lower-complexity reduced models. In this work, we analyze the case in which the reduced model is replaced by a neural surrogate rather than evaluated through a classical numerical scheme.

High-Order Asymptotic-Preserving Schemes for Kinetic Equations from Rarefied to Incompressible Regimes

Giacomo Dimarco, Axel Klar, Theresa Köfler, Lorenzo Pareschi, Tiwari Sudarshan (preprint arXiv:2606.28040)

This work introduces a novel high-order numerical framework for solving kinetic equations, designed to remain uniformly valid across all regimes of the mean free path, spanning from the rarefied kinetic scale to the incompressible hydrodynamic limit. The method is built upon a micro-macro decomposition, which reformulates the underlying kinetic equation into a coupled system consisting of a macroscopic part, representing the fluid-dynamic evolution, and a microscopic part, describing the non-equilibrium deviations. The proposed framework ensures high-order temporal accuracy through the use of Implicit-Explicit Runge-Kutta methods, which provide stability and efficiency in stiff regimes, while spatial resolution is enhanced by combining finite-difference WENO reconstructions with high-order central difference approximations.

How opinions shape epidemics: a graphon-based kinetic approach

Abu Safyan Ali, Elisa Calzola, Giacomo Dimarco, Lorenzo Pareschi, Thomas Rey (preprint arXiv:2605.14139)

Understanding the mutual influence between social behavior and physical health is crucial for designing effective epidemic mitigation strategies. Individual interactions drive the evolution of opinions, which in turn shape how infectious diseases are perceived and consequently how they spread within a population, for instance through the adoption or rejection of preventive measures. At the same time, the distribution and dynamics of physical contacts play a fundamental role in determining transmission patterns.

High-Order Asymptotic-Preserving IMEX schemes for an ES-BGK model for Gas Mixtures

Domenico Caparello, Lorenzo Pareschi, Thomas Rey (preprint arXiv:2511.22304)

In this work we construct a high-order Asymptotic-Preserving (AP) Implicit-Explicit (IMEX) scheme for the ES-BGK model for gas mixtures introduced in [Brull, Commun. Math. Sci., 2015]. The time discretization is based on the IMEX strategy proposed in [Filbet, Jin, J. Sci. Comput., 2011] for the single-species BGK model and is here extended to the multi-species ES-BGK setting. The resulting method is fully explicit, uniformly stable with respect to the Knudsen number and, in the fluid regime, it reduces to a consistent and high-order accurate solver for the limiting macroscopic equations of the mixture.

Swarm-based optimization with jumps: a kinetic BGK framework and convergence analysis

 

Giacomo Borghi, Hyesung Im, Lorenzo Pareschi (to appear in Comm. Pure Appl. Analysis. Preprint arXiv:2507.00871)

Metaheuristic algorithms are powerful tools for global optimization, particularly for non-convex and non-differentiable problems where exact methods are often impractical. Particle-based optimization methods, inspired by swarm intelligence principles, have shown effectiveness due to their ability to balance exploration and exploitation within the search space. In this work, we introduce a novel particle-based optimization algorithm where velocities are updated via random jumps, a strategy commonly used to enhance stochastic exploration.

A data augmentation strategy for deep neural networks with application to epidemic modelling

M. Awais, A.S. Ali, G. Dimarco, F. Ferrarese, L. Pareschi (Bollettino dell'Unione Matematica Italiana
10.1007/s40574-025-00486-3, preprint arXiv:2502.21033)

In this work, we integrate the predictive capabilities of compartmental disease dynamics models with machine learning ability to analyze complex, high-dimensional data and uncover patterns that conventional models may overlook. Specifically, we present a proof of concept demonstrating the application of data-driven methods and deep neural networks to a recently introduced SIR-type model with social features, including a saturated incidence rate, to improve epidemic prediction and forecasting. Our results show that a robust data augmentation strategy trough suitable data-driven models can improve the reliability of Feed-Forward Neural Networks (FNNs) and Nonlinear Autoregressive Networks (NARs), providing a complementary strategy to Physics-Informed Neural Networks, particularly in settings where data augmentation from mechanistic models can enhance learning.

Multiscale constitutive framework of 1D blood flow modeling: asymptotic limits and numerical methods

Giulia Bertaglia, Lorenzo Pareschi (SIAM Multiscale Modeling and Simulation 21(3), 1237-126, 2023. Preprint arXiv:2302.09374)

In this paper, a multiscale constitutive framework for one-dimensional blood flow modeling is presented and discussed. By analyzing the asymptotic limits of the proposed model, it is shown that different types of blood propagation phenomena in arteries and veins can be described through an appropriate choice of scaling parameters, which are related to distinct characterizations of the fluid-structure interaction mechanism (whether elastic or viscoelastic) that exist between vessel walls and blood flow. In these asymptotic limits, well-known blood flow models from the literature are recovered. Additionally, by analyzing the perturbation of the local elastic equilibrium of the system, a new viscoelastic blood flow model is derived.

Dinamiche sociali ed equazioni alle derivate parziali in ambito epidemiologico

Lorenzo Pareschi, Giuseppe Toscani (Matematica, Cultura e Società - Rivista dell'Unione Matematica Italiana, Serie I, Volume 6, No.3, 2021)

In questo breve sunto divulgativo discuteremo l'importanza delle dinamiche sociali in ambito epidemico e la loro modellizzazione matematica tramite equazioni alle derivate parziali. Presenteremo inizialmente modelli di interazione tra individui in cui le caratteristiche sociali, come l'età degli individui, il numero di contatti sociali e la loro ricchezza economica, giocano un ruolo chiave nella diffusione di un'epidemia. Successivamente, accenneremo a modelli che tengono conto anche di caratteristiche aggiuntive quali la carica virale e le difese immunitarie dell'individuo.

Research activity. Numerical methods for hyperbolic problems

Hyperbolic systems of balance laws pose several challenging mathematical and numerical problems. Numerical difficulties arise mainly in presence of nonlinear fluxes which may originate shocks and stiff source terms which may pose severe time step restrictions. The crucial point is to derive schemes able to capture the relevant structure of the solutions such as singularities, shocks, instabilities without resolving the small space and time scales.

Research activity. Numerical solution of kinetic equations

The numerical solution of Boltzmann-type kinetic equations represents a major computational challenge in rarefied gas dynamics and related fields. Typically this is due to the high dimensionality of the problem and to the presence of different time and/or space scales in near-continuum regimes. The necessity of fast solvers for the kinetic integral operators is then an essential part of any numerical schemes for such problems.

Research activity. Kinetic and mean field modelling

Kinetic equations play a major rule in several applications where the multiscale nature of the phenomena cannot be described by a standard macroscopic approach. They are particularly useful in the study of emergent behaviors in complex systems characterized by the spontaneous formation of spatio-temporal structures as a result of simple local interactions between agents. Complex systems mostly appear in the biological and social contexts but can also be encountered in engineering, physics, chemistry, etc.