不确定性动力学方程的平均场控制变量方法及其在社会经济科学中的应用
Mean-field control variate methods for kinetic equations with uncertainties and applications to socio-economic sciences
摘要：
在本文中，我们扩展了最近推出的多保真度控制变量的不确定性量化的玻尔兹曼方程的动力学模型中产生的多智能体系统的研究的情况下。对于这些现象，不确定性的影响是特别明显的，已经开发了几个模型，其平衡态通常是未知的。特别是，我们的目标是开发有效的数值方法的基础上解决的动力学方程在相空间的直接模拟蒙特卡罗（DSMC）耦合到随机空间中的蒙特卡罗采样。为此，利用相应的平均场近似的知识，我们开发了新的平均场控制变量（MFCV）方法，能够大大减少随机空间中的标准蒙特卡洛采样方法的方差。我们验证了这些观察与经典模型的基础上，包括财富交换和意见形成模型的集体现象的几个数值例子。
英文摘要：
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.
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