Monday, June 20, 2022

Locally Structure-Preserving div-curl operators for high order Discontinuous Galerkin schemes

Walter Boscheri, Giacomo Dimarco, Lorenzo Pareschi (preprint arXiv:2206.08609

We propose a novel Structure-Preserving Discontinuous Galerkin (SPDG) operator that recovers at the discrete level the algebraic property related to the divergence of the curl of a vector field, which is typically referred to as div-curl problem. A staggered Cartesian grid is adopted in 3D, where the vector field is naturally defined at the corners of the control volume, while its curl is evaluated as a cell-centered quantity. Firstly, the curl operator is rewritten as the divergence of a tensor, hence allowing compatible finite difference schemes to be devised and to be proven to mimic the algebraic div-curl property.

Wednesday, March 30, 2022

A consensus-based algorithm for multi-objective optimization and its mean-field description

Giacomo Borghi, Michael Herty, Lorenzo Pareschi (preprint arXiv:2203.16384)

We present a multi-agent algorithm for multi-objective optimization problems, which extends the class of consensus-based optimization methods and relies on a scalarization strategy. The optimization is achieved by a set of interacting agents exploring the search space and attempting to solve all scalar sub-problems simultaneously. We show that those dynamics are described by a mean-field model, which is suitable for a theoretical analysis of the algorithm convergence. Numerical results show the validity of the proposed method.

Wednesday, February 23, 2022

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

 Jonathan Franceschi, Lorenzo Pareschi, Mattia Zanella (preprint arXiv:2202.10809)

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.

Monday, January 10, 2022

Effects of vaccination efficacy on wealth distribution in kinetic epidemic models

Emanuele Bernardi, Lorenzo Pareschi, Giuseppe Toscani, Mattia Zanella (Entropy 24(2):216, 2022)

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.

Saturday, January 8, 2022

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.

Friday, December 3, 2021

Multi-fidelity methods for uncertainty propagation in kinetic equations

Giacomo Dimarco, Liu Liu, Lorenzo Pareschi, Xueyu Zhu (to appear in Panoramas & Synthèses, Société Mathématique de France, preprint arXiv:2112.00932)

The construction of efficient methods for uncertainty quantification in kinetic equations represents a challenge due to the high dimensionality of the models: often the computational costs involved become prohibitive. On the other hand, precisely because of the curse of dimensionality, the construction of simplified models capable of providing approximate solutions at a computationally reduced cost has always represented one of the main research strands in the field of kinetic equations.