{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "d18b6d0c",
   "metadata": {},
   "source": [
    "## Setup\n",
    "\n",
    "First, let's import the necessary libraries and set up the environment."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d33ec4c7",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from matplotlib import cm\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "import seaborn as sns\n",
    "\n",
    "# Set plotting style\n",
    "sns.set_style('whitegrid')\n",
    "plt.rcParams['figure.figsize'] = (12, 8)\n",
    "plt.rcParams['font.size'] = 11\n",
    "\n",
    "print(\"✓ Libraries loaded\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fb55613b",
   "metadata": {},
   "source": [
    "## Example 1: Congestion Game\n",
    "\n",
    "Consider a simple congestion game where agents want to move from an initial distribution to a target distribution, but experience congestion costs.\n",
    "\n",
    "### Problem Formulation\n",
    "\n",
    "- **Hamiltonian**: $H(x,p) = \\frac{1}{2}|p|^2$ (quadratic control cost)\n",
    "- **Running cost**: $f(x,m) = \\lambda m(x,t)$ (congestion penalty)\n",
    "- **Terminal cost**: $g(x,m(T)) = \\frac{1}{2}(x - x_{\\text{target}})^2$\n",
    "- **Initial distribution**: $m_0(x) = \\mathcal{N}(0.3, 0.05^2)$\n",
    "\n",
    "### Numerical Solution\n",
    "\n",
    "We solve this using finite difference methods with:\n",
    "- Spatial domain: $\\Omega = [0, 1]$\n",
    "- Time horizon: $T = 1.0$\n",
    "- Grid: $n_x = 100$, $n_t = 100$\n",
    "- Viscosity: $\\nu = 0.01$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "4e31056d",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Problem parameters\n",
    "nx = 100  # Spatial grid points\n",
    "nt = 100  # Time steps\n",
    "T = 1.0   # Time horizon\n",
    "nu = 0.01 # Viscosity\n",
    "lambda_congestion = 0.5  # Congestion penalty\n",
    "x_target = 0.7  # Target location\n",
    "\n",
    "# Spatial and temporal grids\n",
    "x = np.linspace(0, 1, nx)\n",
    "t = np.linspace(0, T, nt)\n",
    "dx = x[1] - x[0]\n",
    "dt = t[1] - t[0]\n",
    "\n",
    "# Initial distribution: Gaussian centered at 0.3\n",
    "m0 = np.exp(-((x - 0.3)**2) / (2 * 0.05**2))\n",
    "m0 /= np.sum(m0) * dx  # Normalize\n",
    "\n",
    "# Plot initial distribution\n",
    "plt.figure(figsize=(10, 4))\n",
    "plt.plot(x, m0, 'b-', linewidth=2, label='Initial distribution $m_0(x)$')\n",
    "plt.axvline(x_target, color='r', linestyle='--', label=f'Target: $x={x_target}$')\n",
    "plt.xlabel('Space $x$')\n",
    "plt.ylabel('Density')\n",
    "plt.title('Initial Agent Distribution')\n",
    "plt.legend()\n",
    "plt.grid(True, alpha=0.3)\n",
    "plt.tight_layout()\n",
    "plt.show()\n",
    "\n",
    "print(f\"Grid: {nx} × {nt}\")\n",
    "print(f\"dx = {dx:.4f}, dt = {dt:.4f}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dd49d343",
   "metadata": {},
   "source": [
    "### Fixed-Point Iteration Algorithm\n",
    "\n",
    "The algorithm proceeds as follows:\n",
    "\n",
    "1. **Initialize**: Start with uniform distribution $m^{(0)}(x,t) = 1$\n",
    "2. **Iterate** until convergence:\n",
    "   - Solve HJB backward: $-\\partial_t u^{(k)} - \\nu \\Delta u^{(k)} + \\frac{1}{2}|\\nabla u^{(k)}|^2 = \\lambda m^{(k-1)}$\n",
    "   - Solve FP forward: $\\partial_t m^{(k)} - \\nu \\Delta m^{(k)} - \\text{div}(m^{(k)} \\nabla u^{(k)}) = 0$\n",
    "   - Update: $m^{(k)} \\leftarrow \\alpha m^{(k)} + (1-\\alpha) m^{(k-1)}$ (relaxation)\n",
    "3. **Check** convergence: $\\|m^{(k)} - m^{(k-1)}\\|_{L^2} < \\epsilon$\n",
    "\n",
    "This is implemented using:\n",
    "- Upwind finite differences for first derivatives\n",
    "- Central differences for second derivatives\n",
    "- Implicit time stepping for stability"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b421e735",
   "metadata": {},
   "outputs": [],
   "source": [
    "def solve_hjb(m, u_T):\n",
    "    \"\"\"Solve HJB equation backward in time\"\"\"\n",
    "    u = np.zeros((nx, nt))\n",
    "    u[:, -1] = u_T  # Terminal condition\n",
    "    \n",
    "    for n in range(nt-2, -1, -1):\n",
    "        for i in range(1, nx-1):\n",
    "            # Laplacian (central difference)\n",
    "            u_xx = (u[i+1, n+1] - 2*u[i, n+1] + u[i-1, n+1]) / dx**2\n",
    "            \n",
    "            # Hamiltonian with upwind scheme\n",
    "            u_x_plus = (u[i+1, n+1] - u[i, n+1]) / dx\n",
    "            u_x_minus = (u[i, n+1] - u[i-1, n+1]) / dx\n",
    "            H = 0.5 * min(u_x_plus**2, u_x_minus**2)  # Upwind\n",
    "            \n",
    "            # Running cost\n",
    "            f = lambda_congestion * m[i, n]\n",
    "            \n",
    "            # Update (implicit Euler)\n",
    "            u[i, n] = u[i, n+1] - dt * (nu * u_xx - H + f)\n",
    "        \n",
    "        # Boundary conditions (Neumann)\n",
    "        u[0, n] = u[1, n]\n",
    "        u[-1, n] = u[-2, n]\n",
    "    \n",
    "    return u\n",
    "\n",
    "def solve_fp(u, m0):\n",
    "    \"\"\"Solve Fokker-Planck equation forward in time\"\"\"\n",
    "    m = np.zeros((nx, nt))\n",
    "    m[:, 0] = m0  # Initial condition\n",
    "    \n",
    "    for n in range(nt-1):\n",
    "        for i in range(1, nx-1):\n",
    "            # Laplacian\n",
    "            m_xx = (m[i+1, n] - 2*m[i, n] + m[i-1, n]) / dx**2\n",
    "            \n",
    "            # Velocity field\n",
    "            u_x = (u[i+1, n] - u[i-1, n]) / (2*dx)\n",
    "            v = u_x  # For quadratic Hamiltonian: H_p = p\n",
    "            \n",
    "            # Upwind for advection\n",
    "            if v > 0:\n",
    "                flux_diff = v * (m[i, n] - m[i-1, n]) / dx\n",
    "            else:\n",
    "                flux_diff = v * (m[i+1, n] - m[i, n]) / dx\n",
    "            \n",
    "            # Update (forward Euler)\n",
    "            m[i, n+1] = m[i, n] + dt * (nu * m_xx - flux_diff)\n",
    "            m[i, n+1] = max(m[i, n+1], 0)  # Non-negativity\n",
    "        \n",
    "        # Boundary conditions\n",
    "        m[0, n+1] = m[1, n+1]\n",
    "        m[-1, n+1] = m[-2, n+1]\n",
    "        \n",
    "        # Normalize\n",
    "        m[:, n+1] /= (np.sum(m[:, n+1]) * dx)\n",
    "    \n",
    "    return m\n",
    "\n",
    "print(\"✓ Solver functions defined\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6ffb4f23",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Fixed-point iteration\n",
    "max_iter = 50\n",
    "tol = 1e-5\n",
    "relax = 0.5\n",
    "\n",
    "# Initialize\n",
    "m_old = np.ones((nx, nt)) / nx\n",
    "errors = []\n",
    "\n",
    "print(\"Running fixed-point iteration...\")\n",
    "for iter in range(max_iter):\n",
    "    # Terminal condition for HJB\n",
    "    u_T = 0.5 * (x - x_target)**2\n",
    "    \n",
    "    # Solve HJB backward\n",
    "    u = solve_hjb(m_old, u_T)\n",
    "    \n",
    "    # Solve FP forward\n",
    "    m_new = solve_fp(u, m0)\n",
    "    \n",
    "    # Check convergence\n",
    "    error = np.sqrt(np.sum((m_new - m_old)**2)) / np.sqrt(np.sum(m_old**2))\n",
    "    errors.append(error)\n",
    "    \n",
    "    if iter % 5 == 0:\n",
    "        print(f\"  Iteration {iter:3d}: error = {error:.6f}\")\n",
    "    \n",
    "    if error < tol:\n",
    "        print(f\"✓ Converged in {iter+1} iterations\")\n",
    "        break\n",
    "    \n",
    "    # Relaxation\n",
    "    m_old = relax * m_new + (1 - relax) * m_old\n",
    "\n",
    "# Plot convergence\n",
    "plt.figure(figsize=(10, 4))\n",
    "plt.semilogy(errors, 'b-', linewidth=2)\n",
    "plt.axhline(tol, color='r', linestyle='--', label=f'Tolerance: {tol}')\n",
    "plt.xlabel('Iteration')\n",
    "plt.ylabel('Relative L² error')\n",
    "plt.title('Convergence of Fixed-Point Iteration')\n",
    "plt.legend()\n",
    "plt.grid(True, alpha=0.3)\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "be038cc7",
   "metadata": {},
   "source": [
    "### Results Visualization\n",
    "\n",
    "Now let's visualize the solution: value function $u(x,t)$ and distribution $m(x,t)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "09bf4fae",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Create meshgrid for plotting\n",
    "X, T = np.meshgrid(x, t)\n",
    "\n",
    "# Plot distribution evolution\n",
    "fig = plt.figure(figsize=(16, 6))\n",
    "\n",
    "# 3D surface plot of distribution\n",
    "ax1 = fig.add_subplot(121, projection='3d')\n",
    "surf1 = ax1.plot_surface(X, T, m_new.T, cmap=cm.viridis, alpha=0.8)\n",
    "ax1.set_xlabel('Space $x$')\n",
    "ax1.set_ylabel('Time $t$')\n",
    "ax1.set_zlabel('Density $m(x,t)$')\n",
    "ax1.set_title('Distribution Evolution')\n",
    "fig.colorbar(surf1, ax=ax1, shrink=0.5)\n",
    "\n",
    "# 3D surface plot of value function\n",
    "ax2 = fig.add_subplot(122, projection='3d')\n",
    "surf2 = ax2.plot_surface(X, T, u.T, cmap=cm.plasma, alpha=0.8)\n",
    "ax2.set_xlabel('Space $x$')\n",
    "ax2.set_ylabel('Time $t$')\n",
    "ax2.set_zlabel('Value $u(x,t)$')\n",
    "ax2.set_title('Value Function')\n",
    "fig.colorbar(surf2, ax=ax2, shrink=0.5)\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "2489123c",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Temporal snapshots\n",
    "fig, axes = plt.subplots(1, 3, figsize=(15, 4))\n",
    "time_indices = [0, nt//2, nt-1]\n",
    "times = [0.0, T/2, T]\n",
    "\n",
    "for ax, idx, time_val in zip(axes, time_indices, times):\n",
    "    ax.plot(x, m_new[:, idx], 'b-', linewidth=2, label='Distribution')\n",
    "    ax.axvline(x_target, color='r', linestyle='--', alpha=0.5, label='Target')\n",
    "    ax.set_xlabel('Space $x$')\n",
    "    ax.set_ylabel('Density')\n",
    "    ax.set_title(f'$m(x, t={time_val:.1f})$')\n",
    "    ax.legend()\n",
    "    ax.grid(True, alpha=0.3)\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()\n",
    "\n",
    "print(\"✓ Solution computed and visualized\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "07bdc875",
   "metadata": {},
   "source": [
    "### Analysis\n",
    "\n",
    "From the results, we observe:\n",
    "\n",
    "1. **Agent Migration**: Agents move from initial position (0.3) toward target (0.7)\n",
    "2. **Congestion Effect**: The distribution spreads out due to congestion penalty $\\lambda m$\n",
    "3. **Nash Equilibrium**: The solution represents a Nash equilibrium where no agent can improve their cost by deviating\n",
    "4. **Value Function**: Shows the optimal cost-to-go from each position at each time\n",
    "\n",
    "The numerical method successfully captures:\n",
    "- Mass conservation: $\\int_\\Omega m(x,t)\\,dx = 1$ for all $t$\n",
    "- Non-negativity: $m(x,t) \\geq 0$\n",
    "- Convergence to equilibrium within 30-50 iterations"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a8f7500e",
   "metadata": {},
   "source": [
    "## Conclusion\n",
    "\n",
    "This notebook demonstrated:\n",
    "- Mathematical formulation of Mean Field Games\n",
    "- Numerical solution via finite difference methods\n",
    "- Fixed-point iteration for coupled HJB-FP system\n",
    "- Visualization and analysis of equilibrium solutions\n",
    "\n",
    "### Further Extensions\n",
    "\n",
    "The `optimizr` Rust library provides additional capabilities:\n",
    "- **Primal-dual methods** for faster convergence\n",
    "- **Higher dimensions** (2D, 3D spatial domains)\n",
    "- **Non-quadratic Hamiltonians**\n",
    "- **State constraints** and obstacle problems\n",
    "- **Parallel computation** for large-scale problems\n",
    "\n",
    "### Related Methods\n",
    "\n",
    "For mean-field variational inference via optimal transport, see the work by Jiang, Chewi, and Pooladian (2023), which develops polyhedral optimization methods in the Wasserstein space for computing mean-field approximations.\n",
    "\n",
    "---\n",
    "\n",
    "**Next Steps:**\n",
    "- Example 2: Multi-population games\n",
    "- Example 3: Mean field type control\n",
    "- Example 4: Comparison with particle methods"
   ]
  }
 ],
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