MEAN-FIELD CONTROL VARIATE METHODS FOR KINETIC EQUATIONS WITH UNCERTAINTIES AND APPLICATIONS TO SOCIOECONOMIC SCIENCES

International Journal for Uncertainty Quantification, vol. 12, pp. 61–84

Abstract

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. To this end, exploiting the knowledge of the corresponding mean-field approximation we develop novel mean-field Control Variate (MFCV) 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 opinion formation model for collective phenomena.

Authors 3

1. Lorenzo Pareschi

   University of Ferrara

   Affiliation as printed

   Department of Mathematics and Computer Science, University of Ferrara, Italy

   University of Ferrara

2. Torsten Trimborn Aachen Institut für Geometrie und Praktische Mathematik

   RWTH Aachen University

   Affiliation as printed

   Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Germany

   RWTH Aachen University

3. Mattia Zanella

   University of Pavia

   Affiliation as printed

   Department of Mathematics "F. Casorati," University of Pavia, Italy

   University of Pavia