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

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 coupled to a Monte Carlo sampling in the random space. To this end, by exploiting the knowledge of the corresponding mean-field approximation we develop novel mean field control variate methods that are able to strongly reduce the variance of the standard Monte Carlo sampling method in the random space. We verify these observations with several numerical examples based on classical models, including wealth exchanges and the opinion formation model for collective phenomena.