Anisotropic Diffusion in Consensus-based Optimization on the Sphere

Massimo Fornasier, Hui Huang, Lorenzo Pareschi, Philippe S√ľnnen (1/4/2021 preprint arXiv:2104.00420)

In this paper we are concerned with the global minimization of a possibly non-smooth and non-convex objective function constrained on the unit hypersphere by means of a multi-agent derivative-free method. The proposed algorithm falls into the class of the recently introduced Consensus-Based Optimization. In fact, agents move on the sphere driven by a drift towards an instantaneous consensus point, which is computed as a convex combination of agent locations, weighted by the cost function according to Laplaces principle, and it represents an approximation to a global minimizer. The dynamics is further perturbed by an anisotropic random vector field to favor exploration.

Mean-field control variate methods for kinetic equations with uncertainties and applications to socio-economic sciences


Lorenzo Pareschi, Torsten Trimborn, Mattia Zanella (4/2/2021 preprint arXiv:2102.02589

In this paper, we extend a recently introduced multi-fidelity control variate for the uncertainty quantification of the Boltzmann equation to the case of kinetic models arising in the study of multiagent systems. For these phenomena, where the effect of uncertainties is particularly evident, several models have been developed whose equilibrium states are typically unknown. In particular, we aim to develop efficient numerical methods based on solving the kinetic equations in the phase space by Direct Simulation Monte Carlo (DSMC) coupled to a Monte Carlo sampling in the random space.

Modeling and simulating the spatial spread of an epidemic through multiscale kinetic transport equations


Walter Boscheri, Giacomo Dimarco, Lorenzo Pareschi (18/12/2020, preprint arXiv:2012.10101, to appear in Math. Mod. Meth. App. Scie.)

In this work we propose a novel space-dependent multiscale model for the spread of infectious diseases in a two-dimensional spatial context on realistic geographical scenarios. The model couples a system of kinetic transport equations describing a population of commuters moving on a large scale (extra-urban) with a system of diffusion equations characterizing the non commuting population acting over a small scale (urban). The modeling approach permits to avoid unrealistic effects of traditional diffusion models in epidemiology, like infinite propagation speed on large scales and mass migration dynamics. A construction based on the transport formalism of kinetic theory allows to give a clear model interpretation to the interactions between infected and susceptible in compartmental space-dependent models.

From particle swarm optimization to consensus based optimization: stochastic modeling and mean-field limit


Sara Grassi, Lorenzo Pareschi (10/12/2020 preprint arXiv:2012.05613), Math. Mod. Meth. App. Scie. to appear

In this paper we consider a continuous description based on stochastic differential equations of the popular particle swarm optimization (PSO) process for solving global optimization problems and derive in the large particle limit the corresponding mean-field approximation based on Vlasov-Fokker-Planck-type equations. The disadvantage of memory effects induced by the need to store the local best position is overcome by the introduction of an additional differential equation describing the evolution of the local best.

On the stability of equilibrium preserving spectral methods for the homogeneous Boltzmann equation

 Lorenzo Pareschi, Thomas Rey (11/11/2020 preprint arXiv:2011.05811, to appear in Applied Math. Letters)

Spectral methods, thanks to the high accuracy and the possibility to use fast algorithms, represent an effective way to approximate the Boltzmann collision operator. On the other hand, the loss of some local invariants leads to the wrong long time behavior. A way to overcome this drawback, without sacrificing spectral accuracy, has been proposed recently with the construction of equilibrium preserving spectral methods. Despite the ability to capture the steady state with arbitrary accuracy, the theoretical properties of the method have never been studied in details. In this paper, using the perturbation argument developed by Filbet and Mouhot for the homogeneous Boltzmann equation, we prove stability, convergence and spectrally accurate long time behavior of the equilibrium preserving approach.