Thursday, October 12, 2023

Kinetic description and convergence analysis of genetic algorithms for global optimization

Giacomo Borghi, Lorenzo Pareschi (preprint arXiv:2310.08562)

Genetic Algorithms (GA) are a class of metaheuristic global optimization methods inspired by the process of natural selection among individuals in a population. Despite their widespread use, a comprehensive theoretical analysis of these methods remains challenging due to the complexity of the heuristic mechanisms involved. In this work, relying on the tools of statistical physics, we take a first step towards a mathematical understanding of GA by showing how their behavior for a large number of individuals can be approximated through a time-discrete kinetic model. This allows us to prove the convergence of the algorithm towards a global minimum under mild assumptions on the objective function for a popular choice of selection mechanism. Furthermore, we derive a time-continuous model of GA, represented by a Boltzmann-like partial differential equation, and establish relations with other kinetic and mean-field dynamics in optimization.

Saturday, August 5, 2023

Gradient-based Monte Carlo methods for relaxation approximations of hyperbolic conservation laws

Giulia Bertaglia, Lorenzo Pareschi, Russel E. Caflisch (preprint arXiv:2308.02904)

Particle methods based on evolving the spatial derivatives of the solution were originally introduced to simulate reaction-diffusion processes, inspired by vortex methods for the Navier--Stokes equations. Such methods, referred to as gradient random walk methods, were extensively studied in the '90s and have several interesting features, such as being grid free, automatically adapting to the solution by concentrating elements where the gradient is large and significantly reducing the variance of the standard random walk approach. In this work, we revive these ideas by showing how to generalize the approach to a larger class of partial differential equations, including hyperbolic systems of conservation laws.

Wednesday, June 14, 2023

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

Andrea Medaglia, Lorenzo Pareschi, Mattia Zanella (preprint arXiv:2306.07701)

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.

Saturday, February 18, 2023

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.

Thursday, February 2, 2023

Modeling opinion polarization on social media: application to Covid-19 vaccination hesitancy in Italy

Jonathan Franceschi, Lorenzo Pareschi, Elena Bellodi, Marco Gavanelli, Marco Bresadola (PLoS ONE18(10):e0291993, 2023)

The SARS-CoV-2 pandemic reminded us how vaccination can be a divisive topic on which the public conversation is permeated by misleading claims, and thoughts tend to polarize, especially on online social networks. In this work, motivated by recent natural language processing techniques to systematically extract and quantify opinions from text messages, we present a differential framework for bivariate opinion formation dynamics that is coupled with a compartmental model for fake news dissemination.