Equation-Free Analysis of Macroscopic Behavior in Traffic and Pedestrian Flow

Abstract

Equation-free methods make possible an analysis of the evolution of a few coarse-grained or macroscopic quantities for a detailed and realistic model with a large number of fine-grained or microscopic variables, even though no equations are explicitly given on the macroscopic level. This will facilitate a study of how the model behavior depends on parameter values including an understanding of transitions between different types of qualitative behavior. These methods are introduced and explained for traffic jam formation and emergence of oscillatory pedestrian counter flow in a corridor with a narrow door.

Pattern formation in flows of asymmetrically interacting particles: Peristaltic pedestrian dynamics as a case study

“…Equation (34) can be easily solved, by applying the Fourier transform (23), and its solution has the form…”

“…Furthermore, an extension to more complicated geometries, e.g., annuli [15,26] and buildings, might be an interesting case for the asymmetric model. Finally, the macroscopic dynamics of the pedestrian flow is worth studying with multiscale methods such as diffusion maps [22] or equation-free methods [19,23,24,5]. A macroscopic analysis might find the observed transitions between regimes in a numerically simple way and could shed some light on the type of bifurcations by using a coarse bifurcation analysis.…”

Bifurcation of learning and structure formation in neuronal maps

“…As we will see, under the high-dimensional microscopic neuronal and synaptic dynamics a sound localization map arises that can be reduced to a bifurcation of a corresponding low-dimensional order parameter. The bifurcation analysis of the macroscopic dynamics is made possible by equation-free methods [2,[9][10][11]. The surprise is both in the type of bifurcation and in the ensuing map that behaves as a slowly traveling wave.…”

“…The computation time of the traveling wave dynamics can be significantly reduced by using a coarse projective integration method, i.e., an Euler scheme with time step Δt for the macroscopic dynamics of eq. (11). The speed-up depends basically on the required accuracy and is determined by the ratio Δt/δt; cf.…”

Convergence of Equation-Free Methods in the Case of Finite Time Scale Separation with Application to Deterministic and Stochastic Systems

“…Classical applications of equation-free methods were macroscopic bifurcation analysis for microscopic simulations in chemical engineering (see [24] for a review). Recently similar analysis was performed on stochastic network models of neurons [4,30] or disease spread [18], or on agent-based models in ecology [45] and social sciences (for example, for consumer lock-in [3], for pedestrian flow [33,32], or for trading [42]). Another example for a high-level task accessible via equation-free methods is control design [43,42].…”