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Data driven learning to enhance a kinetic model of distressed crowd dynamics

Daewa Kim, Demetrio Labate, Kamrun Mily, Annalisa Quaini

TL;DR

This paper considers a kinetic model of crowd dynamics that features the level of stress as a parameter and proposes to estimate this key parameter by solving an inverse crowd dynamics problem by solving a forward crowd dynamics problem.

Abstract

The mathematical modeling of crowds is complicated by the fact that crowds possess the behavioral ability to develop and adapt moving strategies in response to the context. For example, in emergency situations, people tend to alter their walking strategy in response to fear. To be able to simulate these situations, we consider a kinetic model of crowd dynamics that features the level of stress as a parameter and propose to estimate this key parameter by solving an inverse crowd dynamics problem. This paper states the mathematical problem and presents a method for its numerical solution. We show some preliminary results based on a synthetic data set, i.e., test cases where the exact stress level is known and the crowd density data are generated numerically by solving a forward crowd dynamics problem.

Data driven learning to enhance a kinetic model of distressed crowd dynamics

TL;DR

This paper considers a kinetic model of crowd dynamics that features the level of stress as a parameter and proposes to estimate this key parameter by solving an inverse crowd dynamics problem by solving a forward crowd dynamics problem.

Abstract

The mathematical modeling of crowds is complicated by the fact that crowds possess the behavioral ability to develop and adapt moving strategies in response to the context. For example, in emergency situations, people tend to alter their walking strategy in response to fear. To be able to simulate these situations, we consider a kinetic model of crowd dynamics that features the level of stress as a parameter and propose to estimate this key parameter by solving an inverse crowd dynamics problem. This paper states the mathematical problem and presents a method for its numerical solution. We show some preliminary results based on a synthetic data set, i.e., test cases where the exact stress level is known and the crowd density data are generated numerically by solving a forward crowd dynamics problem.

Paper Structure

This paper contains 14 sections, 38 equations, 18 figures.

Table of Contents

  1. Introduction
  2. The time continuous inverse crowd dynamics problem
  3. The forward problem
  4. The time-continuous inverse problem
  5. The time discrete inverse crowd dynamics problem
  6. The time discrete forward problem
  7. The time discrete inverse problem and the KKT conditions
  8. Numerical solution of the inverse problem
  9. Space discretization
  10. Numerical results
  11. Circular chamber without partial obstruction near exit
  12. Circular chamber with partial obstruction
  13. Square chamber with exit at the corner of the walls
  14. Conclusions and perspectives

Figures (18)

  • Figure 1: Security camera frames of a surge in the crowd gathered in Torino (Italy) to watch a soccer game on June 3rd, 2017. The stampede led to 3 people dead and over 1600 injured. Video available at https://youtu.be/yuqcNgcgzIA?feature=shared.
  • Figure 2: Snapshots of experiments with panicking ants from Ref. SHIWAKOTI20111433: circular chamber without (left) and with (right) column obstructing the exit.
  • Figure 3: Dependence of the dimensionless velocity modulus $v$ on the dimensionless density $\rho$. The dashed red line connects the origin of the axis to the critical value $\rho_c = 1/5$.
  • Figure 4: Sketch of computational domain $\Omega$ with exit $E$ and a person located at ${\boldsymbol{x}}$, walking with direction $\theta_h$. The person should choose direction ${\boldsymbol{u}}_E$ to reach the exit and direction ${\boldsymbol{u}}_W$ is to avoid collision with the wall. The person's distances form the exit and from the wall where they would collide are $d_E$ and $d_W$, respectively.
  • Figure 5: Left: Computational domain for the circular chamber without column, with initial density and direction for the numerical simulation. Right: snapshot from repetition 2 out of 30.
  • ...and 13 more figures

Theorems & Definitions (2)

  • Remark 2.1
  • Remark 2.2