README.md

Solution of a Mean-Field Game's Stationary State with Discount Factor

This software allows for the solution of the stationary state of the Mean-Field Games model described in publications Bonnemain et al. and Butano et al., modelling the passage through a dense crowd of a cylindrical obstacle at constant speed $s$. The equations we want to solve are

$$0 =\frac{\sigma^2}{2}\Delta{u^e} - \frac{1}{2\mu}(\vec{\nabla}u^e)^2 - \gamma u^e(x) - V[m^e]$$

$$0 =\frac{\sigma^2}{2}\Delta m^e +\frac{1}{\mu}\nabla\cdot(m^e\nabla u^e)$$

We do this by using a double formulation, using the Cole-Hopf change of variable allowing an elegant connection to the theory of the Non-Linear Schrödinger Equation.

Install and Use the Module

To use this python module, first of all create a directory named "project_mfg". Inside this folder, create the following directories

• data
• gfx
• configs

Then, create the python script "run.py", which contains the instructions

• import mfg_ergodic_intruder.mfg_ergodic as mfg : to import the module.
• m = mfg.mfg('config_name', 'mode') : where 'config_name' is the name of a configuration that should be formatted like this.
• m.simulation() : to launch the simulation. Options in the docu.
• m.draw_density() : to draw the MFG density. Options in the docu.
• m.draw_velocities() : to draw the MFG velocities. Options in the docu.