In this paper we develop a framework for parameter estimation in macroscopic pedestrian models using individual trajectories -- microscopic data. We consider a unidirectional flow of pedestrians in a corridor and assume that the velocity decreases with the average density according to the fundamental diagram. Our model is formed from a coupling between a density dependent stochastic differential equation and a nonlinear partial differential equation for the density, and is hence of McKean--Vlasov type. We discuss identifiability of the parameters appearing in the fundamental diagram from trajectories of individuals, and we introduce optimization and Bayesian methods to perform the identification. We analyze the performance of the developed methodologies in various situations, such as for different in- and outflow conditions, for varying numbers of individual trajectories and for differing channel geometries.
Experiments with pedestrians revealed that the geometry of the domain, as well as the incentive of pedestrians to reach a target as fast as possible have a strong influence on the overall dynamics. In this paper, we propose and validate different mathematical models at the micro- and macroscopic levels to study the influence of both effects. We calibrate the models with experimental data and compare the results at the micro -- as well as macroscopic levels. Our numerical simulations reproduce qualitative experimental features on both levels, and indicate how geometry and motivation level influ…
We study the homogenization of a novel microscopic tridomain system, allowing for a more detailed analysis of the properties of cardiac conduction than the classical bidomain and monodomain models. In (Acta Appl.Math. 179 (2022) 1--35), we detail this model in which gap junctions are considered as the connections between adjacent cells in cardiac muscle and could serve as alternative or supporting pathways for cell-to-cell electrical signal propagation. Departing from this microscopic cellular model, we apply the periodic unfolding method to derive the macroscopic tridomain model. Several diff…
From microscopic theory to macroscopic theory: a systematic study on static modeling for liquid crystals
LMAM & School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）
From Microscopic Damage to Macroscopic Games: A Dimensionality Reduction of Stem Cell Homeostasis
College of Engineering, Boston University（波士顿大学工程学院）；Institute for Global Sustainability, Boston University（波士顿大学全球可持续发展研究所）；Department of Mathematics, Northeastern University（东北大学数学系）；Committee on Computational and Applied Mathematics, University of Chicago（芝加哥大学计算与应用数学委员会）
Effective macroscopic dynamics of stochastic partial differential equations in perforated domains
本文导出了随机微观系统的有效宏观模型。原始的微观系统是由一个随机的偏微分方程定义的域上穿孔的小孔或异质性建模。均匀化的有效模型仍然是一个随机偏微分方程，但定义在一个统一的区域上没有洞。微观模型的解决方案是收敛到有效的宏观模型的概率分布，孔的尺寸减小到零。此外，还证明了宏观系统在\emph{convergence in probability distribution}意义下的长期有效性和宏观系统在\emph{convergence in energy}意义下的有效性.
