Tuesday, April 30, 2024

New trends on the systems approach to modeling SARS-CoV-2 pandemics in a globally connected planet

Giulia Bertaglia, Andrea Bondesan, Diletta Burini, Raluca Eftimie, Lorenzo Pareschi, Giuseppe Toscani (Math. Mod. Meth. App. Sci. to appear. Preprint arXiv:2405.00541)

This paper presents a critical analysis of the literature and perspective research ideas for modeling the epidemics caused by the SARS-CoV-2 virus. It goes beyond deterministic population dynamics to consider several key complexity features of the system under consideration. In particular, the multiscale features of the dynamics from contagion to the subsequent dynamics of competition between the immune system and the proliferating virus. Other topics addressed in this work include the propagation of epidemics in a territory, taking into account local transportation networks, the heterogeneity of the population, and the study of social and economic problems in populations involved in the spread of epidemics.

Friday, February 9, 2024

Conservative polynomial approximations and applications to Fokker-Planck equations


Tino Laidin, Lorenzo Pareschi (preprint arXiv:2402.06473)

We address the problem of constructing approximations based on orthogonal polynomials that preserve an arbitrary set of moments of a given function without loosing the spectral convergence property. To this aim, we compute the constrained polynomial of best approximation for a generic basis of orthogonal polynomials. The construction is entirely general and allows us to derive structure preserving numerical methods for partial differential equations that require the conservation of some moments of the solution, typically representing relevant physical quantities of the problem.

Wednesday, January 10, 2024

Optimization by linear kinetic equations and mean-field Langevin dynamics

Lorenzo Pareschi (preprint arXiv:2401.05553)

Probably one of the most striking examples of the close connections between global optimization processes and statistical physics is the simulated annealing method, inspired by the famous Monte Carlo algorithm devised by Metropolis et al. in the middle of the last century. In this paper we show how the tools of linear kinetic theory allow to describe this gradient-free algorithm from the perspective of statistical physics and how convergence to the global minimum can be related to classical entropy inequalities. This analysis highlight the strong link between linear Boltzmann equations and stochastic optimization methods governed by Markov processes.

Sunday, December 31, 2023

Reduced variance random batch methods for nonlocal PDEs

Lorenzo Pareschi, Mattia Zanella (to appear in Acta Applicandae Mathematicae. Preprint arXiv:2401.00493)

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.

Modelling contagious viral dynamics: a kinetic approach based on mutual utility

Giulia Bertaglia, Lorenzo Pareschi, Giuseppe Toscani (Math. Biosci. Eng. 21(3), 4241-4268, 2024. Preprint arXiv:2401.00480)

The time evolution of a contagious viral disease is modeled as the dynamic progression of different classes of populations that interact pairwise, aiming to improve their condition with respect to a given target. This evolutionary mechanism is based on binary interactions between agents, designed to enhance their utility mutually. To achieve this goal, we introduce kinetic equations of Boltzmann type to describe the time evolution of the probability distributions of the agent system undergoing binary interactions. The fundamental idea is to describe these interactions using principles from price theory, particularly by employing Cobb-Douglas utility functions for the binary exchange and the Edgeworth box to depict the common exchange area where utility increases for both agents.