Constrained consensus-based optimization

Giacomo Borghi, Michael Herty, Lorenzo Pareschi (23/11/21 preprint arXiv:2111.10571)

In this work we are interested in the construction of numerical methods for high dimensional constrained nonlinear optimization problems by particle-based gradient-free techniques. A consensus-based optimization (CBO) approach combined with suitable penalization techniques is introduced for this purpose. The method relies on a reformulation of the constrained minimization problem in an unconstrained problem for a penalty function and extends to the constrained settings the class of CBO methods. Exact penalization is employed and, since the optimal penalty parameter is unknown, an iterative strategy is proposed that successively updates the parameter based on the constrained violation.

Bi-fidelity stochastic collocation methods for epidemic transport models with uncertainties

Giulia Bertaglia, Liu Liu, Lorenzo Pareschi, Xueyu Zhu (28/10/2021 preprint arXiv:2110.14579)

Uncertainty in data is certainly one of the main problems in epidemiology, as shown by the recent COVID-19 pandemic. The need for efficient methods capable of quantifying uncertainty in the mathematical model is essential in order to produce realistic scenarios of the spread of infection. In this paper, we introduce a bi-fidelity approach to quantify uncertainty in spatially dependent epidemic models. The approach is based on evaluating a high-fidelity model on a small number of samples properly selected from a large number of evaluations of a low-fidelity model.

Kinetic modelling of epidemic dynamics: social contacts, control with uncertain data, and multiscale spatial dynamics

Giacomo Albi, Giulia Bertaglia, Walter Boscheri, Giacomo Dimarco, Lorenzo Pareschi, Giuseppe Toscani, Mattia Zanella (4/10/2021 to appear in Predicting Pandemics in a Globally Connected World, Vol. 1, N. Bellomo and M. Chaplain Editors, Springer-Nature (2021). Preprint arXiv:2110.00293)

In this survey we report some recent results in the mathematical modeling of epidemic phenomena through the use of kinetic equations. We initially consider models of interaction between agents in which social characteristics play a key role in the spread of an epidemic, such as the age of individuals, the number of social contacts, and their economic wealth. Subsequently, for such models, we discuss the possibility of containing the epidemic through an appropriate optimal control formulation based on the policy maker's perception of the progress of the epidemic. The role of uncertainty in the data is also discussed and addressed.

Spreading of fake news, competence, and learning: kinetic modeling and numerical approximation

 Jonathan Franceschi, Lorenzo Pareschi (28/9/2021 to appear in Philosophical Transactions of the Royal Society A. Preprint arXiv:2109.14087)

The rise of social networks as the primary means of communication in almost every country in the world has simultaneously triggered an increase in the amount of fake news circulating online. This fact became particularly evident during the 2016 U.S. political elections and even more so with the advent of the COVID-19 pandemic. Several research studies have shown how the effects of fake news dissemination can be mitigated by promoting greater competence through lifelong learning and discussion communities, and generally rigorous training in the scientific method and broad interdisciplinary education. The urgent need for models that can describe the growing infodemic of fake news has been highlighted by the current pandemic.

Mean-field particle swarm optimization

 

Sara Grassi, Hui Huang, Lorenzo Pareschi, Jinniao Qiu (3/8/2021 to appear in Modeling and Simulation for Collective Dynamics, IMS Lecture Note Series, World Scientific, preprint arXiv:2108.00393)

In this work we survey some recent results on the global minimization of a non-convex and possibly non-smooth high dimensional objective function by means of particle based gradient-free methods. Such problems arise in many situations of contemporary interest in machine learning and signal processing. After a brief overview of metaheuristic methods based on particle swarm optimization (PSO), we introduce a continuous formulation via second-order systems of stochastic differential equations that generalize PSO methods and provide the basis for their theoretical analysis. Subsequently, we will show how through the use of mean-field techniques it is possible to derive in the limit of large particles number the corresponding mean-field PSO description based on Vlasov-Fokker-Planck type equations.