Mean field games model a variety of social phenomena such as crowd motion. In
R. Andreev, Preconditioning the augmented Lagrangian method for instationary mean field games with diffusion, HAL report, 2017 (Link).
we developed space-time preconditioners for finite element discretizations of mean field games.

The following simulation shows a crowd moving from one room to another while avoiding an obstacle. The MATLAB code for this simulation is available as examples/mfg2d in the ppfem package.

The density $\rho$ of the crowd is assumed to evolve according to $\partial_t \rho - \nu^2 \Delta \rho + \nabla \cdot (\rho \mathbf{v}) = 0$ where $\mathbf{v}$ is the velocity field and $\nu$ is a diffusion constant subsuming random and untracked effects. The velocity field is determined by minimizing the functional $\int_{\text{space-time}} \{ L(\mathbf{v}) \rho + A(\rho) \} \text{d}x \text{d}t + \int_{\text{space}} \Gamma(\rho|_{\text{final time}}) \text{d}x$ that consists of transport energy and friction (i.e. crowd discomfort), and a term that models the desired location of the crowd at a certain final time.

The following images show the difference in the resulting behavior when increasing the diffusion coefficient

from $\nu = 0.1$:
to $\nu = 1$:

This work was supported by the French ANR-12-MONU-0013 "ISOTACE" and the Swiss NSF 164616.

R. Andreev, 2018-03-26.