宏观人群流动模型的建模与仿真
Modeling and Simulation of Macroscopic Pedestrian Flow Models
AI总结：
本文分析了行人运动的宏观模型，如Hughes模型和具有非线性移动性的均场博弈模型，通过最优控制方法推导出均场博弈结构，并提供了数值结果以展示两种模型在单维和二维中的关系。
摘要：
我们数值分析了一些宏观行人运动模型，如Hughes模型[1]和具有非线性移动性的均场博弈模型[2]，用于模拟人群快速撤离场景。Hughes模型由一个非线性守恒定律描述行人密度，结合一个势方程来建模任务中的常识。通过最优控制方法得到均场博弈模型，其中每个行人的运动由最小化一个成本函数决定，该函数依赖于位置、速度、撤离时间和总体人群密度。我们考虑了非线性移动性的抛物线最优控制问题，导致均场博弈结构。我们展示了与Hughes模型相关的最优控制问题。此外，我们提供了几种数值结果，将两种模型联系在一起。
英文摘要：
We analyze numerically some macroscopic models of pedestrian motion such as Hughes model [1] and mean field game with nonlinear mobilities [2] modeling fast exit scenarios in pedestrian crowds. A model introduced by Hughes consisting of a non-linear conservation law for the density of pedestrians coupled with an Eikonal equation for a potential modeling the common sense of the task. Mean field game with nonlinear mobilities is obtained by an optimal control approach, where the motion of every pedestrian is determined by minimizing a cost functional, which depends on the position, velocity, exit time and the overall density of people. We consider a parabolic optimal control problem of nonlinear mobility in pedestrian dynamics, which leads to a mean field game structure. We show how optimal control problem related to the Hughes model for pedestrian motion. Furthermore we provide several numerical results which relate both models in one and two dimensions. References [1] Hughes R.L.: A continuum theory for the flow of pedestrians, Transportation Research Part B: Methodological, 36, 507-535 (2000) [2] Burger M., Di Francesco M., Markowich P.A., Wolfram M-T.: Mean field games with nonlinear mobilities in pedestrian dynamics, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 19, 1311-1333 (2014)
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