Hierarchical dynamic domain decomposition for the multiscale Boltzmann equation

Domenico Caparello, Lorenzo Pareschi, Thomas Rey (preprint arXiv:2505.03360)

In this work, we present a hierarchical domain decomposition method for the multi-scale Boltzmann equation based on moment realizability matrices, a concept introduced by Levermore, Morokoff, and Nadiga. This criterion is used to dynamically partition the two-dimensional spatial domain into three regimes: the Euler regime, an intermediate kinetic regime governed by the ES-BGK model, and the full Boltzmann regime. The key advantage of this approach lies in the use of Euler equations in regions where the flow is near hydrodynamic equilibrium, the ES-BGK model in moderately non-equilibrium regions where a fluid description is insufficient but full kinetic resolution is not yet necessary, and the full Boltzmann solver where strong non-equilibrium effects dominate, such as near shocks and boundary layers. This allows for both high accuracy and significant computational savings, as the Euler solver and the ES-BGK models are considerably cheaper than the full kinetic Boltzmann model.

Wasserstein convergence rates for stochastic particle approximation of Boltzmann models

 G. Borghi, L. Pareschi (preprint arXiv: 2504.10091). 

We establish quantitative convergence rates for stochastic particle approximation based on Nanbu-type Monte Carlo schemes applied to a broad class of collisional kinetic models. Using coupling techniques and stability estimates in the Wasserstein-1 (Kantorovich-Rubinstein) metric, we derive sharp error bounds that reflect the nonlinear interaction structure of the models. Our framework includes classical Nanbu Monte Carlo method and more recent developments as Time Relaxed Monte Carlo methods. The results bridge the gap between probabilistic particle approximations and deterministic numerical error analysis, and provide a unified perspective for the convergence theory of Monte Carlo methods for Boltzmann-type equations. As a by-product, we also obtain existence and uniqueness of solutions to a large class of Boltzmann-type equations.

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

M. Awais, A.S. Ali, G. Dimarco, F. Ferrarese, L. Pareschi (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.

Multi-fidelity and multi-level Monte Carlo methods for kinetic models of traffic flow

Elisa Iacomini, Lorenzo Pareschi (preprint arXiv:2501.15967)

In traffic flow modeling, incorporating uncertainty is crucial for accurately capturing the complexities of real-world scenarios. In this work we focus on kinetic models of traffic flow, where a key step is to design effective numerical tools for analyzing uncertainties in vehicles interactions. To this end we discuss space-homogeneous Boltzmann-type equations, employing a non intrusive Monte Carlo approach both on the physical space, to solve the kinetic equation, and on the stochastic space, to investigate the uncertainty. To address the high dimensional challenges posed by this coupling, control variate approaches such as multi-fidelity and multi-level Monte Carlo methods are particularly effective.

Implicit-Explicit Methods for Evolutionary Partial Differential Equations

Sebastiano Boscarino, Lorenzo Pareschi, and Giovanni Russo, SIAM Mathematical Modelling and Computations Series, 2024 

Publishing December 2024

Implicit-explicit (IMEX) time discretization methods have proven to be highly effective for the numerical solution of a wide class of evolutionary partial differential equations (PDEs) across various contexts. These methods have become mainstream for solving evolutionary PDEs, particularly in the fields of hyperbolic and kinetic equations. The first book on the subject, Implicit-Explicit Methods for Evolutionary Partial Differential Equations provides an in-depth yet accessible approach. The authors summarize and illustrate the construction, analysis, and application of IMEX methods using examples, test cases, and implementation details; guide readers through the various methods and teach them how to select and use the one most appropriate for their needs; and demonstrate how to identify stiff terms and effectively implement high-order methods in time for a variety of systems of PDEs. Readers interested in learning modern techniques for the effective numerical solution of evolutionary PDEs with multiple time scales will find in this book a unified, compact, and accessible treatment.