Time-Evolving Measures and Macroscopic Modeling of Pedestrian Flow

Piccoli, Benedetto; Tosin, Andrea · 2010 · Crossref

DOI: 10.1007/s00205-010-0366-y

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Summary

This paper introduces a macroscopic model for pedestrian flow based on a measure-theoretical framework, addressing the limitations of existing microscopic and hyperbolic conservation law approaches. While microscopic models accurately describe individual agents but struggle with global system analysis, and standard macroscopic models rely on fluid dynamics analogies that introduce technical difficulties in handling boundaries and multidimensional interactions, this work proposes a discrete-time Eulerian representation. The model describes space occupancy via a family of measures pushed forward by motion mappings, facilitating the treatment of nonlocal interactions, obstacles, and wall boundary conditions. The methodology defines the evolution of pedestrian density through a sequence of positive measures $\mu_n$ on a spatial domain $\Omega$. The dynamics are governed by a push-forward operation $\mu_{n+1} = \gamma_n \# \mu_n$, where $\gamma_n$ is a motion mapping derived from a velocity field $v_n$ and a time step $\Delta t$. This discrete-time approach serves as an explicit discretization of a continuous-time conservation law. The authors establish the theoretical well-posedness of the model, proving that if the initial measure is absolutely continuous with respect to the Lebesgue measure, this property is preserved over time. Furthermore, they derive bounds for the $L^1$ and $L^\infty$ norms of the density functions, ensuring stability. A spatial approximation scheme is developed using piecewise constant functions on nested grids, allowing for numerical implementation with convergence guarantees as the grid resolution increases. The primary findings demonstrate that the measure-theoretical approach provides a robust mathematical structure for crowd dynamics. The authors prove that the total mass (number of pedestrians) is conserved within the domain, provided the motion mappings keep the support of the measure within $\Omega$. The theoretical analysis confirms that the discrete push-forward operation correctly approximates the continuous conservation law under specific regularity conditions on the velocity field. The derived numerical scheme allows for the computation of density evolution without the analytical complexities associated with solving multidimensional hyperbolic partial differential equations. The significance of this work lies in offering an alternative macroscopic modeling tool that avoids the technical pitfalls of traditional hyperbolic conservation laws, particularly regarding boundary conditions and multidimensional analysis. By leveraging measure theory, the model naturally accommodates nonlocal interactions and complex geometries, such as obstacles and bounded domains. This framework is particularly suitable for applications in engineering and social sciences requiring the management and optimization of pedestrian fluxes in crowded environments like airports and stadiums. The paper establishes a foundation for further research into control and optimization problems within crowd dynamics, providing a mathematically rigorous yet computationally tractable approach to simulating pedestrian flow.

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• pedestrian behavior perception