Multi-element probabilistic collocation solution for dynamic continuum pedestrian models with random inputs

Abstract

This study focuses on dynamic continuum pedestrian flow models with random inputs, which can be represented by sets of partial differential equations with some modeling parameters being randomized. Under random conditions, the model outputs are no longer fixed and may differ appreciably from their respective average levels. Simulating the resulting distribution is important as it helps quantify the effects of uncertainties on traffic behaviors when evaluating walking facilities. Through two examples based on continuum models, the effect of random inputs on pedestrian flow propagation is qualitatively analyzed. Crowd evacuation is found to be effective in reducing the variation and risk produced by randomness, while congestion is observed to significantly increase the uncertainty within the system. For a general system without an explicitly known exact solution, an existing efficient solver — the multi-element probabilistic collocation method (ME-PCM) — is introduced to derive the solution distribution numerically. The ME-PCM is non-intrusive and flexible and has no limitations in terms of governing partial differential equations and the numerical schemes for solving them. The ME-PCM’s use of element-wise local orthogonal polynomials to represent the solution enables it to converge efficiently even if shocks occur during the modeling period. As a demonstration case, the well-known Hughes model is applied in a numerical example with a corridor and an obstacle. The demand at the inflow boundary is randomized to a lognormal distribution that represents day-to-day demand stochasticity. The results indicate that the ME-PCM’s solution converges more rapidly than those of the Monte Carlo and generalized polynomial chaos methods. Statistical information on pedestrian density is derived from the ME-PCM solution and can be used to identify the locations in walking facilities where the average pedestrian density is moderate but where exceptional congestion with a large variance can occur. This successful application shows the possibility of quantifying the uncertainty in pedestrian flow models using the ME-PCM. The proposed approach can also be applied to models with other similar random inputs, given that a well-established algorithm for deterministic cases is available.