{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Mean field game solver for fast exit problems\n", "\n", "We consider a mean field game system describing the evolution of density $\\rho$, which wants to leave a domain as fast as possible. The corresponding macroscopic optimal control problem is given by\n", "\\begin{align*}\n", "J(\\rho,v) = \\frac{1}{2} \\int_0^T \\int_{\\Omega} F(\\rho) \\lvert v \\rvert^2 dx dt + \\frac{1}{2} \\int_0^T \\int_{\\Omega} E(\\rho) dx dy\n", "\\end{align*}\n", "subject to the constraint that $\\partial_t \\rho(x,t) = \\frac{\\sigma^2}{2} \\Delta \\rho - \\nabla \\cdot (G(\\rho) v)$. Here the function $G$ corresponds to a nonlinear mobility, for example $G(\\rho) = \\rho_{\\max} - \\rho$, $F(\\rho)$ an increased cost of motion in the case of high denities. The function $E$ may penalize large densities.\\\\\n", "We consider the problem on a bounded domain, with Robin type boundary conditions for $\\rho$ as well as $\\phi$. Let $j = -\\frac{\\sigma^2}{2} \\nabla \\rho + G(\\rho) v$ denote the total flux, and $n$ the unit normal outer vector. Then we set\n", "\\begin{align*}\n", "j \\cdot n &= \\beta \\rho \\\\\n", "\\nabla \\phi \\cdot n &= -\\beta \\phi \n", "\\end{align*}\n", "at the exits.\n", "\n", "The solver is based on a fixed point iteration for the optimality system:\n", "