Prediction-Correction Pedestrian Flow by Means of Minimum Flow Problem

TL;DR

This paper introduces a novel mathematical model for predicting pedestrian flow that combines a transport equation with a minimum flow problem, accommodating various interactions and boundary conditions, validated through numerical simulations.

Contribution

The paper develops a new prediction-correction model for crowd dynamics using an eikonal equation and a minimum flow problem, adaptable to different interaction types and boundary conditions.

Findings

Model effectively captures diverse pedestrian behaviors.

Numerical simulations demonstrate flexibility and robustness.

Applicable to various boundary conditions and interaction models.

Abstract

We study a new variant of mathematical prediction-correction model for crowd motion. The prediction phase is handled by a transport equation where the vector field is computed via an eikonal equation $\Vert \nabla\varphi\Vert=f$, with a positive continuous function $f$ connected to the speed of the spontaneous travel. The correction phase is handled by a new version of the minimum flow problem. This model is flexible and can take into account different types of interactions between the agents, from gradient flow in Wassersetin space to granular type dynamics like in sandpile. Furthermore, different boundary conditions can be used, such as non-homogeneous Dirichlet (e.g., outings with different exit-cost penalty) and Neumann boundary conditions (e.g., entrances with different rates). Combining finite volume method for the transport equation and Chambolle-Pock's primal dual algorithm for the eikonal equation and minimum flow problem, we present numerical simulations to demonstrate the behavior in different scenarios.