01_K.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 1.\n",
    "### Sensitivity of hydraulic head at a point to <font color='blue'>spatially uniform hydraulic conductivity</font> under steady state flow conditions\n",
    "\n",
    "## 0. Forward model\n",
    "<br>\n",
    "Governing equation:\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad K\\ b\\ \\frac{d^2h}{dx^2}+R=0\\cr\n",
    "\\cr\\end{align}$\n",
    "\n",
    "Boundary conditions:\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad -K\\ b\\ \\frac{dh(x)}{dx}=0\\ ,\\quad\\quad\\;\\;\\, x=0=\\Gamma_2\\cr\n",
    "\\end{align}$\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad h(x)=h_{\\Gamma_1}\\ ,\\quad\\quad\\quad\\quad\\quad x=L=\\Gamma_1\\cr\n",
    "\\cr\\end{align}$\n",
    "\n",
    "Closed-form solution:\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad h(x)=h_L+\\frac{R (L^2-x^2)}{2\\ K\\ b}\n",
    "\\cr\\cr\\end{align}$\n",
    "\n",
    "Spatial derivatives of hydraulic head obtained from differentiation:\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad \\frac{d h}{d x}=-\\frac{R\\ x}{K\\ b},\\quad\\quad \\frac{d^2 h}{d x^2}{}=-\\frac{R}{K\\ b}\n",
    "\\cr\\end{align}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from IPython.display import HTML, display\n",
    "def set_background(color):\n",
    "    script = (\n",
    "        \"var cell = this.closest('.code_cell');\"\n",
    "        \"var editor = cell.querySelector('.input_area');\"\n",
    "        \"editor.style.background='{}';\"\n",
    "        \"this.parentNode.removeChild(this)\").format(color)\n",
    "    display(HTML('<img src onerror=\"{}\">'.format(script)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from warnings import filterwarnings\n",
    "filterwarnings(\"ignore\", category=DeprecationWarning)\n",
    "\n",
    "import numpy as np\n",
    "\n",
    "def h(x, K, R, b, L, BC1h):\n",
    "    return R/K/2./b*(L**2.-x**2.)\n",
    "\n",
    "K, R, b, L, BC1h, ocol = 10., 1e-1/1000., 10., 10000., 0., 5000\n",
    "X = np.arange(L)\n",
    "H0 = np.array([h(x, K, R, b, L, BC1h) for x in X])\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "plt.subplots(figsize=[8,2])\n",
    "plt.plot(X/1000., H0, 'k--', mfc='none', label='forward solution, analytical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel('$h$ (m)')\n",
    "plt.legend(loc=3);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Direct sensitivity\n",
    "$\\begin{align}\\cr\n",
    "\\frac{\\partial h(x')}{\\partial K}=\\frac{R\\ (x^2-L^2)}{2\\ K^2\\ b}\n",
    "\\cr\\cr\\end{align}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "dhdK = [R/K**2./2./b*(x**2.-L**2.) for x in X]\n",
    "benchmark = dhdK[ocol]\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "plt.subplots(figsize=[8,2])\n",
    "plt.plot(X/1000., dhdK, 'k--', mfc='none', label='direct sensitivity, analytical')\n",
    "plt.plot(ocol/1000., dhdK[ocol], 'ro')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\partial h / \\partial K$')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<img src onerror=\"var cell = this.closest('.code_cell');var editor = cell.querySelector('.input_area');editor.style.background='rgba(0, 200, 0, 0.2)';this.parentNode.removeChild(this)\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-3.750000\n"
     ]
    }
   ],
   "source": [
    "set_background('rgba(0, 200, 0, 0.2)')\n",
    "print('%.6f'% benchmark)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Perturbation sensitivity\n",
    "$\\begin{align}\\cr\n",
    "\\frac{\\partial h(x')}{\\partial K}\\approx\\frac{h(x,K+\\Delta K)-h(x,K)}{\\Delta K}\\;\n",
    "\\cr\\cr\\end{align}$\n",
    "\n",
    "### 2a. Analytical"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "dpar = 1e-4\n",
    "H0 = np.array([h(x, K, R, b, L, BC1h) for x in X])\n",
    "H1 = np.array([h(x, K+K*dpar, R, b, L, BC1h) for x in X])\n",
    "dhdK = (H1-H0)/(K*dpar)\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "plt.subplots(figsize=[8,2])\n",
    "plt.plot(X/1000., dhdK, 'k--', mfc='none', label='perturbation sensitivity, analytical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\partial h / \\partial K$')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<img src onerror=\"var cell = this.closest('.code_cell');var editor = cell.querySelector('.input_area');editor.style.background='rgba(0, 200, 0, 0.2)';this.parentNode.removeChild(this)\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-3.749625 (-3.750000)\n"
     ]
    }
   ],
   "source": [
    "set_background('rgba(0, 200, 0, 0.2)')\n",
    "print('%.6f (%.6f)'% (dhdK[ocol], benchmark))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2b. Semi-analytical"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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k4OXlpQld3TZrJfEYEQmwamS3QZO4Uqoqy8nJ4YcffqBHjx4A+Pn5kZaWxrhx45gwYQJ9+/bVRVpUmVjrmvgeY4xObaSUUiVwdna2JHAR4c9//jOhoaGsXr2aAQMG0LJlS1asWGHjKFV1UZYkHgxEG2OOGmMOGWNijDGHKiowpZSyd8YYxowZw8cff0xaWhrr168nJCTEsvJaUlISM2bMICoqSu9DV7elLN3p3jfbrreYKaXU7dm4cSMPPvggeXl5dOjQgYceeoiHHnqIO+64w9ahqSqkXNfEjTFGfuWg0hxjLZrElVLVyc8//8yGDRtYt24d//znP3F0dCQ1NRUPDw/y8/P1+rkq/wIoxphpxpjW1xVa2xgzwBjzITDJGoEqpVRN07BhQ6ZMmUJkZCRJSUl89NFHeHh4ADBs2DAGDx7M6tWruXjxoo0jVVVRaZL4ECAf+NgYc9oY84Mx5iRwnML7xheJyKoKjFEppWqEVq1aMWHCBKBwUFyvXr04duwYkyZNomnTpjzwwAPs2rXLxlGqqqQ0C6DkiMgyEelD4bzpoUAPEfEWkSkiEl3RQSqlVE1jjGHu3LmcPHmS3bt3M3nyZLZt28a+ffsAuHTpEvv27dMBcTVcmWbwF5E8ETkjIufL8jpjTCtjzA5jTJwx5rAx5n+KtnsYY741xhwv+rdhWcpVSqnqzhhDnz59WLp0KWfOnGHq1KlA4aC4O++8Ez8/P+bPn8+pU6dsHKmyhcpahucq8HsR6QT0Bp4puud8FrBdRNpROJ3rrEqKRyml7I6TkxOurq4AjB49mvfeew8PDw9mzZpF69atGTx4MJcuXbJxlKoyVUoSL2q9Hyj6+SIQB7QE7gU+LDrsQ2B0ZcSjlFL2zs3NjSlTprB7926OHz/O7NmzcXFxoV69egB88MEH7Nq1S7vbq7my3Cc+AJgInAdigUNArIhcKel1NynHB9gJ+ANJIuJebN/PIlJil7reYqaUUiXLy8ujVatWpKSk4Ovry6OPPsqjjz5KmzZtbB2aug3WmnZ1LfAlsAdoA/yJwsVQyhJIPWAj8JyIXCjD6540xkQZY6LS0tLKckqllKpxnJycOHHiBKtXr6ZNmzbMnTuXtm3b8s4779g6NGVlZWmJ7xSRvrd9ImOcKPwS8A8Reato21Ggn4icMcY0ByJFpENJ5WhLXCmlyiYpKYm1a9cyZswYOnXqxM6dO1m3bh1PPPEEQUFBGGNsHaIqQbla4saY1caY54B/G2N+f5sBGOADIO6XBF5kM/+dKGYS8PntlK+UUurWWrduzUsvvUSnTp0AiIuLY82aNdx555107dqVxYsXk5GRYeMo1e0ozbSroUDXYo+GwH+KHodEZP2vnsSYYGAXEMN/1yJ/CdgLhAOtgSRgvIicK6ksbYkrpVT5ZWZm8ve//53333+fqKgofHx8OHnypLbKqyCrrCderLBaQGcKE3oXEXm+/CGWniZxpZSyrv/85z8kJSUxcuRI8vPz6d+/P2FhYUyePJlWrVrZOrwaz1oD2wAQkasickhE1lR2AldKKWV9Xbt2ZeTIkQCkp6dTp04dXnnlFby9vRk6dCgbN24kNzfXxlGqm6msyV6UUkrZgaZNm/Ltt99y8uRJ/vjHPxITE8N9991HREQEgN53XsVoEldKKXUDX19f5s6dS2JiIl9//TUDBw4EYPbs2QwdOpTPP/+cq1ev2jhKpUlcKaXULTk6OjJ48GDLuuZNmjTh0KFDjB49Gh8fH1599VWdt92GNIkrpZQqteeee47ExEQ2bdqEv78/c+bM4Q9/+INlf0FBQQmvVtamSVwppVSZ1KpVi9GjR/P1119z4sQJ5s6dC0BsbCxt27bljTfeICUlxcZR1gyaxJVSSt22Nm3a0K5dOwCuXLmCr68vL730El5eXkyYMIHIyEgdDFeBNIkrpZSyisDAQCIiIoiLi2PatGls27aN4cOHc+FC4VIZmsytT5O4Ukopq+rYsSNvvfUWp06d4ptvvsHNzQ0RITQ0lOnTp3P8+HFbh1htaBJXSilVIVxcXOjTpw8A2dnZNG/enKVLl9K+fXuGDBnCl19+SX5+vo2jtG+axJVSSlU4V1dX1q1bR1JSEnPnziUmJoaRI0fyySef2Do0u6ZJXCmlVKVp1qwZL7/8MgkJCYSHhzNmzBgAli9fzhNPPEF0dLRtA7QzmsSVUkpVOicnJ8aPH4+LiwsAaWlpfPTRR3Tv3p3g4GDCw8N1RrhS0CSulFLK5l5++WVOnTrFW2+9xdmzZ7n//vuZOHGircOq8jSJK6WUqhIaNmzI9OnTOXr0KJs3b2batGkAnDlzhmnTpnHs2DEbR1j1aBJXSilVpTg6OjJy5EiCg4MB+Pe//817771Hhw4dGDlyJNu3b9d7zotoEldKKVWljR07lqSkJF555RX27t1LWFgYPXr04MqVK7YOzeaMvX2bCQoKkqioqGu25eXlkZycTE5Ojo2iUqrqcXZ2xsvLCycnJ1uHopTV5OTk8PHHH3P06FHmzZsHwJo1axg4cCDNmjWzcXQVwxizX0SCbrqvOiTx+Ph46tevT6NGjTDG2CgypaoOESEjI4OLFy/i6+tr63CUqjCnT5+mVatWODo68uCDDzJ9+nS6detm67CsqqQkXi2603NycjSBK1WMMYZGjRpp75Sq9lq0aMGRI0eYOnUqGzdupHv37oSFhdWYQXDVIokDmsCVuo5+JlRN0a5dOxYvXkxycjLz588nMTERDw8PAJKSksjNzbVxhBWn2iRxW1u8eDGdOnWqEvc1zpkzh4ULF5arjMjISEaMGFHiMefPn2fZsmWW56dPn+a+++4r13kry5/+9Ce2bdtWoecozf/D22+/zeXLly3Phw0bxvnz58t8rlWrVvHss8+W+XVKVSfu7u7MnDmTY8eO4enpiYgwfvx42rRpw8KFCy2rqVUnmsStZNmyZWzdupV169aV6nhrzUQkIhQUFFilrLK6Pom3aNGCDRs22CSWspo7dy5hYWG2DuOGJL5161bc3d1tF5BS1UDxXqg5c+bQvn17nn/+eVq1asXMmTNJTk62YXTWpUncCqZOncrJkycZNWoUixYt4ty5c4wePZouXbrQu3dvDh06BBT+Mj355JMMGjSIRx99lGHDhln2de/enblz5wKFMxe9//77XLp0idDQUHr06EFAQACff/45AAkJCXTq1Imnn36aHj168NNPP/H666/ToUMHwsLCOHr06E3jXL9+Pf7+/nTt2pW+ffsCheMJJk+eTEBAAN27d2fHjh03vO76FqW/vz8JCQnMmjWLEydO0K1bN55//nkSEhLw9/cvsdxVq1YxduxYhgwZQrt27Zg5c+av1u/ixYvp3LkzXbp04YEHHgAgKyuLxx9/nJ49e9K9e3dL3axatYrRo0czcuRIfH19eeedd3jrrbfo3r07vXv35ty5cwA89thjN/3C8Wt1PmXKFPz8/Bg0aBDZ2dkArFixgp49e9K1a1fGjRt3TVIGOHHiBD169LA8P378OIGBgSxevJjTp0/Tv39/+vfvD4CPjw/p6ekArF69mi5dutC1a1ceeeQRAL744gt69eplue6XkpLyq/WnVE1ljGHo0KFERESwb98+hg0bxl//+le++OILoJqsby4idvUIDAyU6/3www/XPL/nnntueCxdulRERLKysm66f+XKlSIikpaWdsO+0vD29pa0tDQREXn22Wdlzpw5IiKyfft26dq1q4iIvPLKK9KjRw+5fPmyiIi88cYb8s4770hmZqYEBQXJoEGDRESkX79+cuTIEcnLy5PMzExLXG3btpWCggKJj48XY4z8+9//FhGRqKgo8ff3l6ysLMnMzJS2bdvKggULbojR399fkpOTRUTk559/FhGRhQsXymOPPSYiInFxcdKqVSvJzs6WHTt2yPDhwy1xFy/Pz89P4uPjJT4+Xvz8/Czbiz+/VbkrV64UX19fOX/+vGRnZ0vr1q0lKSmpxLpt3ry55OTkXBP3iy++KGvWrLFsa9eunVy6dElWrlwpbdu2lQsXLkhqaqo0aNBA3n33XRERee6552TRokUiIjJp0iRZv379Decqqc4dHR3l4MGDIiIyfvx4y/nT09Mtr589e7YsXrz4hnrr16+f5bUvvvii5ZjivzfFn8fGxkr79u0t+zIyMkRE5Ny5c1JQUCAiIitWrJAZM2aIiMjKlSvlmWeeueH9XP/ZUKqmi4+Pl6ysLBERWbZsmQwZMkS2b99u+VxVRUCU3CInaku8AuzevdvSchowYAAZGRlkZmYCMGrUKMuE/yEhIezcuZPdu3czfPhwLl26xOXLl0lISKBDhw6ICC+99BJdunQhLCyMU6dOWVpe3t7e9O7dG4Bdu3YxZswYXF1dadCgAaNGjbppXH369OGxxx5jxYoVljV8i8fasWNHvL29rTKqs6RyQ0NDcXNzw9nZmc6dO5OYmFhiWV26dGHixImsXbuWWrVqAfDNN98wb948unXrRr9+/cjJySEpKQmA/v37U79+fRo3boybmxsjR44EICAggISEhBLPVVKd+/r6Wm5dCQwMtJQVGxtLSEgIAQEBrFu3jsOHD99Q7hNPPMHKlSvJz8/nk08+4aGHHioxjoiICO677z48PT0BLIN0kpOTGTx4MAEBASxYsOCm51JK3ZqPjw+urq5A4SIsBw4cIDQ0lKCgINavX29365vXsnUAFSEyMvKW+1xdXUvc7+npWeL+0pCbdNH8co2mbt26lm09e/YkKiqKNm3aMHDgQNLT01mxYgWBgYEArFu3jrS0NPbv34+TkxM+Pj6WW4aKl1O8/JIsX76cvXv3smXLFrp160Z0dHSpupNq1ap1zXX30ty2VFK5derUsfzs6Oj4q+MDtmzZws6dO9m8eTOvvfYahw8fRkTYuHEjHTp0uObYvXv3XlO+g4OD5bmDg8MN59q7dy9PPfUUUHid/Ny5c7es8+vj/qU7/bHHHuOzzz6ja9eurFq16qa/P+PGjePVV19lwIABBAYG0qhRoxLfs4jc9P902rRpzJgxg1GjRhEZGcmcOXNKLEcpdWtPPPEEDz/8MGvWrGHBggVMmDCB+++/n7///e+2Dq3UtCVeAfr27WsZ4BYZGYmnpycNGjS44bjatWvTqlUrwsPD6d27NyEhISxcuJCQkBAAMjMzadKkCU5OTuzYseOWLda+ffuyadMmsrOzuXjxouV6z/VOnDhBr169mDt3Lp6envz000/XxHrs2DGSkpJuSIw+Pj4cOHAAgAMHDhAfHw9A/fr1uXjx4q/Wwa3Kvd6jjz7K999/f822goICfvrpJ/r378+bb77J+fPnuXTpEoMHD2bJkiWWLwsHDx4ssexb6dWrF9HR0URHRzNq1KhS13lxFy9epHnz5uTl5d1yYKOzszODBw/mt7/9LZMnT7Zsv1UdhoaGEh4eTkZGBoDlWn5mZiYtW7YE4MMPPyzz+1VKXcvZ2ZkpU6YQFxdHeHi45Ut9eno6b7/9NpcuXbJxhCXTJF4B5syZQ1RUFF26dGHWrFkl/rENCQmhadOmuLq6EhISQnJysiWJT5w4kaioKIKCgli3bh0dO3a8aRk9evTg/vvvp1u3bowbN87y+us9//zzBAQE4O/vT9++fenatStPP/00+fn5BAQEcP/997Nq1aprWpxQ2Io8d+4c3bp1491336V9+/YANGrUiD59+uDv78/zzz9/zWtKU+71Dh06RPPmza/Zlp+fz8MPP2wZIDd9+nTc3d15+eWXycvLo0uXLvj7+/Pyyy+XWHZplbbOi3vttdfo1asXAwcOLPH4iRMnYoxh0KBBlm1PPvkkQ4cOtQxs+4Wfnx+zZ8/mnnvuoWvXrsyYMQMo/N0aP348ISEhlq52pVT5OTo6Mn78eMtncdOmTUyfPh1vb29effVVyxfpqqZaTLsaFxdHp06dbBSRsoYLFy7wm9/8hvXr19s6lAqzcOFCMjMzee211yrtnPrZUOr2fffdd8ybN48vvviCunXr8tvf/pb58+fj4FC57d+Spl2tltfElf1p0KBBtU7gY8aM4cSJE0RERNg6FKVUKd19991s3ryZmJgY5s+fT3x8vCWBp6am0qRJExtHWAW6040xQ4wxR40xPxpjZtk6HqUqwqZNmzh06JB2gStlhwICAli7di3h4eFA4VwPXl5ePPjgg/znP/+xaWw2TeLGGEdgKTAU6Aw8aIzpbMuYlFJKqZv5pRXu7u7O9OnTLXf6jBgxgj179tgmJpuc9b/uBH4UkZMikgv8Hbj3dgqyt2v7SlU0/UwoVTEaN27M/PnzSUpK4rXXXmPPnj3069fPcjdJZbJ1Em8J/FTseXLRtjJxdnYmIyND/2gpVUSK1hN3dna2dShKVVvu7u788Y9/JCEhgS+//PJX53+oCLYe2HazGUpuyMTGmCeBJwFat259wwu8vLxITk4mLS3N6gEqZa+cnZ3x8vKydRhKVXv16tWz2YJKtk7iyUCrYs+9gNPXHyQi7wHvQeEtZtfvd3JywtfXt6JiVEoppaokW3en7wPaGWN8jTG1gQeAzTaOSSmllLILNm2Ji8hVY8yzwD8AR+D/RERXdFBKKaVKwdbd6YjIVmCrreNQSiml7I3dTbtqjEkDfn1VitLzBNKtWF5NpHVoHVqP5ad1WH5ah+Vn7Tr0FpHGN9thd0nc2owxUbeak1aVjtahdWg9lp/WYflpHZZfZdahrQe2KaWUUuo2aRJXSiml7JQm8aL7z1W5aB1ah9Zj+Wkdlp/WYflVWh3W+GviSimllL3SlrhSSillp2p0Ete1zMvHGNPKGLPDGBNnjDlsjPkfW8dkr4wxjsaYg8aYL20diz0yxrgbYzYYY44U/T7eZeuY7I0xZnrR5zjWGPOxMUZXzykFY8z/GWNSjTGxxbZ5GGO+NcYcL/q3YUWdv8YmcV3L3CquAr8XkU5Ab+AZrcPb9j9AnK2DsGP/C3wtIh2BrmhdlokxpiXwOyBIRPwpnEHzAdtGZTdWAUOu2zYL2C4i7YDtRc8rRI1N4lhxLfOaSkTOiMiBop8vUviHs8xLydZ0xhgvYDjwvq1jsUfGmAZAX+ADABHJFZHzNg3KPtUCXIwxtQBXbrIYlbqRiOwEzl23+V7gw6KfPwRGV9T5a3ISt8pa5qqQMcYH6A7stXEo9uhtYCZQYOM47FUbIA1YWXRJ4n1jTF1bB2VPROQUsBBIAs4AmSLyjW2jsmtNReQMFDZ2gCYVdaKanMRLtZa5+nXGmHrARuA5Eblg63jsiTFmBJAqIvttHYsdqwX0AN4Vke5AFhXYfVkdFV2zvRfwBVoAdY0xD9s2KlUaNTmJl2otc1UyY4wThQl8nYh8aut47FAfYJQxJoHCSzoDjDFrbRuS3UkGkkXkl16gDRQmdVV6YUC8iKSJSB7wKXC3jWOyZynGmOYARf+mVtSJanIS17XMy8kYYyi8DhknIm/ZOh57JCIvioiXiPhQ+DsYISLaAioDETkL/GSM6VC0KRT4wYYh2aMkoLcxxrXocx2KDg4sj83ApKKfJwGfV9SJbL4Uqa3oWuZW0Qd4BIgxxkQXbXupaHlZpSrTNGBd0Rfyk8BkG8djV0RkrzFmA3CAwrtODqIzt5WKMeZjoB/gaYxJBl4B5gHhxpjfUPgFaXyFnV9nbFNKKaXsU03uTldKKaXsmiZxpZRSyk5pEldKKaXslCZxpZRSyk5pEldKKaXslCZxpZRSyk5pEldKKaXslCZxpdQ1jDEuxph/Fq1x7lN8neQyllPbGLOzaFUspVQF0CSulLre48CnIpJfnkKKlvjdDtxvlaiUUjfQJK5UDWKM2WGMGVj085+NMYtvcthEbjLXszGmTdFSnz2LWuhHipb9jDXGrDPGhBlj/mWMOW6MubPoZZ8VlaeUqgCaxJWqWV4BZhtjJlK4/vv04juL5h5vIyIJ123vQOFqdZNFZF/R5juA/wW6AB2Bh4Bg4A/AS0XHxAI9K+SdKKVq7gIoStVEIrKzaJWqGUC/m3SZewLnr9vWmMKW+bjrFgmKF5EYAGPMYWC7iIgxJgbwKTpfvjEm1xhTX0QuWv8dKVWzaUtcqRrEGBMANAeu3CKpZgPO123LBH6icNW64q4U+7mg2PMCrm0g1AFybjdmpdStaRJXqoYwxjQH1gH3AlnGmMHXHyMiPwOOxpjiiTwXGA08aox5qIznbASkiUjebQeulLolTeJK1QDGGFfgU+D3IhIHvAbMucXh31B4bdtCRLKAEcB0Y8y9ZTh1f0DXl1eqguh64kqpaxhjugMzROQRK5T1KfCiiBwtf2RKqetpS1wpdQ0ROQjsMMY4lqecopHun2kCV6riaEtcKaWUslPaEldKKaXslCZxpZRSyk5pEldKKaXslCZxpZRSyk5pEldKKaXslCZxpZRSyk5pEldKKaXs1P8D/5s52yisJMwAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 576x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from os import getcwd, chdir\n",
    "cwd = getcwd()\n",
    "chdir(r'../../timml')\n",
    "import timml\n",
    "chdir(cwd)\n",
    "\n",
    "M0 = timml.Model3D(kaq=K, z=[0., -b])\n",
    "timml.ImpLineDoublet1D(M0, xld=0.)\n",
    "timml.HeadLineSink1D(M0, xls=L, hls=0.)\n",
    "timml.StripAreaSink(M0, 0., L, R)\n",
    "M0.solve(silent=True)\n",
    "H0 = M0.headalongline(X, 0.).flatten()\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "plt.subplots(figsize=[8,2])\n",
    "plt.plot(X/1000., H0, 'k--', mfc='none', label='forward solution, semi-analytical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel('$h$ (m)')\n",
    "plt.legend(loc=3);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "M1 = timml.Model3D(kaq=K+K*dpar, z=[0., -b])\n",
    "timml.ImpLineDoublet1D(M1, xld=0.)\n",
    "timml.HeadLineSink1D(M1, xls=L, hls=0.)\n",
    "timml.StripAreaSink(M1, 0., L, R)\n",
    "M1.solve(silent=True)\n",
    "H1 = M1.headalongline(X, 0.).flatten()\n",
    "dhdK = (H1-H0)/(K*dpar)\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "plt.subplots(figsize=[8,2])\n",
    "plt.plot(X/1000., dhdK, 'k--', mfc='none', label='perturbation sensitivity, semi-analytical')\n",
    "plt.plot(ocol/1000., dhdK[ocol], 'ro')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\partial h / \\partial K$')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<img src onerror=\"var cell = this.closest('.code_cell');var editor = cell.querySelector('.input_area');editor.style.background='rgba(0, 200, 0, 0.2)';this.parentNode.removeChild(this)\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-3.749625 (-3.750000)\n"
     ]
    }
   ],
   "source": [
    "set_background('rgba(0, 200, 0, 0.2)')\n",
    "print('%.6f (%.6f)'% (dhdK[ocol], benchmark))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2c. Numerical"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import flopy\n",
    "\n",
    "nrow, ncol = 1, int(L)\n",
    "M0 = flopy.modflow.Modflow(modelname='model', exe_name='../mf2005.exe')\n",
    "flopy.modflow.ModflowDis(M0, nlay=1, nrow=nrow, ncol=ncol, nper=1, delr=1., delc=1., top=0., botm=-b, steady=True, \n",
    "                         perlen=1., nstp=1)\n",
    "flopy.modflow.ModflowBas(M0, ibound=np.hstack([np.ones([nrow, ncol-1], dtype=int), -1*np.ones([1,1])]), \n",
    "                         strt=BC1h*np.ones([nrow, ncol], dtype=float))\n",
    "flopy.modflow.ModflowLpf(M0, hk=K, vka=-999., ss=-999., sy=-999., ipakcb=53)\n",
    "flopy.modflow.ModflowRch(M0, nrchop=1, rech=R, ipakcb=53)\n",
    "flopy.modflow.ModflowPcg(M0, hclose=1e-6, rclose=1e-6)\n",
    "flopy.modflow.ModflowOc(M0, stress_period_data={(0,0): ['save head', 'save budget']})\n",
    "M0.write_input()\n",
    "success, buff = M0.run_model(silent=True)\n",
    "H0 = flopy.utils.binaryfile.HeadFile('model.hds').get_data()[0,0,:]\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "plt.subplots(figsize=[8,2])\n",
    "plt.plot(X/1000., H0, 'k--', mfc='none', label='forward solution, numerical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel('$h$ (m)')\n",
    "plt.legend(loc=3);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "M1 = M0\n",
    "flopy.modflow.ModflowLpf(M1, hk=K+K*dpar, vka=-999., ss=-999., sy=-999., ipakcb=53)\n",
    "M1.write_input()\n",
    "success, buff = M1.run_model(silent=True)\n",
    "H1 = flopy.utils.binaryfile.HeadFile('model.hds').get_data()[0,0,:]\n",
    "dhdK = (H1-H0)/(K*dpar)\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "plt.subplots(figsize=[8,2])\n",
    "plt.plot(X/1000., dhdK, 'k--', mfc='none', label='perturbation sensitivity, numerical')\n",
    "plt.plot(ocol/1000., dhdK[ocol], 'ro')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\partial h / \\partial K$')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<img src onerror=\"var cell = this.closest('.code_cell');var editor = cell.querySelector('.input_area');editor.style.background='rgba(0, 200, 0, 0.2)';this.parentNode.removeChild(this)\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-3.753662 (-3.750000)\n"
     ]
    }
   ],
   "source": [
    "set_background('rgba(0, 200, 0, 0.2)')\n",
    "print('%.6f (%.6f)'% (dhdK[ocol], benchmark))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Adjoint sensitivity\n",
    "\n",
    "<br>\n",
    "    \n",
    "$\\begin{align}\n",
    "\\quad\\quad \\frac{\\partial h(x')}{\\partial K}=\\int\\limits_X\\psi^*_1(x)\\ b\\ \\frac{d^2 h(x)}{d x^2}\\; dx\n",
    "\\cr\\end{align}$\n",
    "\n",
    "Governing equation:\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad K\\ b\\ \\frac{d\\psi^*_1}{dx}+\\frac{1}{2\\ K\\ b}\\delta(x-x')=0\\cr\n",
    "\\cr\\end{align}$\n",
    "\n",
    "Boundary conditions: \n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad -K\\ b\\ \\frac{d\\psi^*_1(x)}{dx}=0\\ ,\\quad\\quad\\; x=0=\\Gamma_2\\cr\n",
    "\\end{align}$\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad \\psi^*_1(x)=0\\ ,\\ \\quad\\quad\\quad\\quad\\quad x=L=\\Gamma_1\\cr\n",
    "\\cr\\end{align}$\n",
    "\n",
    "Closed-form solution:\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad\\quad \\psi^*_1(x)=\\frac{1}{2\\ K\\ b}\\left[H\\left(x'-x\\right)\\left(L-x'\\right)+H\\left(x-x'\\right)\\left(L-x\\right)\\right]\n",
    "\\cr\\end{align}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def a(x, xp, K, b, L):\n",
    "    if x>xp:\n",
    "        a = L-x\n",
    "    else:\n",
    "        a = L-xp\n",
    "    return a/K/b\n",
    "A = np.array([a(x, ocol, K, b, L) for x in X])\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "plt.subplots(figsize=[8,2])\n",
    "plt.plot(X/1000., A, 'k--', mfc='none', label='adjoint solution, analytical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\psi^*_1$ (T/$L^2$)')\n",
    "plt.legend(loc=3);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3a. Analytical"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "dhdK = [np.sum(np.array([a(x, xp, K, b, L) for x in X])*b*-R/(K*b)) for xp in X[::1000]]\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "plt.subplots(figsize=[8,2])\n",
    "plt.plot(X[::1000]/1000., dhdK, 'k--', mfc='none', label='adjoint sensitivity, analytical')\n",
    "plt.plot(ocol/1000., dhdK[int(ocol/1000)], 'ro')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\partial h / \\partial K$')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<img src onerror=\"var cell = this.closest('.code_cell');var editor = cell.querySelector('.input_area');editor.style.background='rgba(0, 200, 0, 0.2)';this.parentNode.removeChild(this)\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-3.750250 (-3.750000)\n"
     ]
    }
   ],
   "source": [
    "set_background('rgba(0, 200, 0, 0.2)')\n",
    "print('%.6f (%.6f)'% (dhdK[int(np.where(X[::1000]==float(ocol))[0])], benchmark))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3b. Semi-analytical"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "M0 = timml.Model3D(kaq=K, z=[0., -b])\n",
    "timml.ImpLineDoublet1D(M0, xld=0.)\n",
    "timml.HeadLineSink1D(M0, xls=L, hls=0.)\n",
    "timml.LineSink1D(M0, xls=float(ocol), sigls=-1.)\n",
    "M0.solve(silent=True)\n",
    "A = M0.headalongline(X, 0.).flatten()\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "plt.subplots(figsize=[8,2])\n",
    "plt.plot(X/1000., A, 'k--', mfc='none', label='adjoint solution, semi-analytical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\psi^*_1$ (T/$L^2$)')\n",
    "plt.legend(loc=3);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "dhdK = np.empty(0)\n",
    "for xp in X[::1000]:\n",
    "    M0 = timml.Model3D(kaq=K, z=[0., -b])\n",
    "    timml.ImpLineDoublet1D(M0, xld=0.)\n",
    "    timml.HeadLineSink1D(M0, xls=L, hls=0.)\n",
    "    timml.LineSink1D(M0, xls=xp, sigls=-1.)\n",
    "    M0.solve(silent=True)\n",
    "    A = M0.headalongline(X, 0.).flatten() \n",
    "    dhdK = np.append(dhdK, np.sum(A*b*-R/(K*b)))\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "plt.subplots(figsize=[8,2])\n",
    "plt.plot(X[::1000][1:]/1000., dhdK[1:], 'k--', mfc='none', label='adjoint sensitivity, semi-analytical')\n",
    "plt.plot(ocol/1000., dhdK[int(ocol/1000)], 'ro')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\partial h / \\partial K$')\n",
    "plt.legend(loc=2);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<img src onerror=\"var cell = this.closest('.code_cell');var editor = cell.querySelector('.input_area');editor.style.background='rgba(0, 200, 0, 0.2)';this.parentNode.removeChild(this)\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-3.750250 (-3.750000)\n"
     ]
    }
   ],
   "source": [
    "set_background('rgba(0, 200, 0, 0.2)')\n",
    "print('%.6f (%.6f)'% (dhdK[int(np.where(X[::1000]==float(ocol))[0])], benchmark))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3c. Numerical, continuous"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "M0 = flopy.modflow.Modflow(modelname='model', exe_name='../mf2005.exe')\n",
    "flopy.modflow.ModflowDis(M0, nlay=1, nrow=1, ncol=ncol, nper=1, delr=1., delc=1., top=0., botm=-b, steady=True, \n",
    "                         perlen=1., nstp=1)\n",
    "flopy.modflow.ModflowBas(M0, ibound=np.hstack([np.ones([nrow, ncol-1], dtype=int), -1*np.ones([1,1])]), \n",
    "                         strt=np.zeros([nrow, ncol], dtype=float))\n",
    "flopy.modflow.ModflowLpf(M0, hk=K, vka=-999., ss=-999., sy=-999., ipakcb=53)\n",
    "flopy.modflow.ModflowWel(M0, stress_period_data={0: [0, 0, ocol, 1.]})\n",
    "flopy.modflow.ModflowPcg(M0, hclose=1e-6, rclose=1e-6)\n",
    "flopy.modflow.ModflowOc(M0, stress_period_data={(0,0): ['save head', 'save budget']})\n",
    "M0.write_input()\n",
    "success, buff = M0.run_model(silent=True)\n",
    "A = flopy.utils.binaryfile.HeadFile('model.hds').get_data()[0,0,:]\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "plt.subplots(figsize=[8,2])\n",
    "plt.plot(X/1000., A, 'k--', mfc='none', label='adjoint solution, numerical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\psi^*_1$ (T/$L^2$)')\n",
    "plt.legend(loc=3);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "dhdK = np.empty(0)\n",
    "for oc,xp in enumerate(X[::1000]):\n",
    "    M1 = M0\n",
    "    flopy.modflow.ModflowWel(M1, stress_period_data={0: [0, 0, oc*1000, 1.]})\n",
    "    M1.write_input()\n",
    "    success, buff = M1.run_model(silent=True)\n",
    "    A = flopy.utils.binaryfile.HeadFile('model.hds').get_data()[0,0,:]\n",
    "    dhdK = np.append(dhdK, np.sum(A*b*-R/(K*b)))\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "f,s = plt.subplots(figsize=[8,2])\n",
    "plt.plot(X[::1000]/1000., dhdK, 'k--', mfc='none', label='adjoint sensitivity, numerical')\n",
    "plt.plot(ocol/1000., dhdK[int(ocol/1000)], 'ro')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel(r'$\\partial h / \\partial K$')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<img src onerror=\"var cell = this.closest('.code_cell');var editor = cell.querySelector('.input_area');editor.style.background='rgba(0, 200, 0, 0.2)';this.parentNode.removeChild(this)\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-3.749250 (-3.750000)\n"
     ]
    }
   ],
   "source": [
    "set_background('rgba(0, 200, 0, 0.2)')\n",
    "print('%.6f (%.6f)'% (dhdK[int(np.where(X[::1000]==float(ocol))[0])], benchmark))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3d. Numerical, discrete"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Re-run adjoint model to obtain the adjoint state vector, $\\psi^*_1$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1, 10000)"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M1 = M0\n",
    "flopy.modflow.ModflowWel(M1, stress_period_data={0: [0, 0, ocol, 1.]})\n",
    "M1.write_input()\n",
    "success, buff = M1.run_model(silent=True)\n",
    "psi = np.atleast_2d(flopy.utils.binaryfile.HeadFile('model.hds').get_data()[0,0,:])\n",
    "np.shape(psi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Re-run forward model to obtain vectors of output heads, $h$, cell-by-cell conductances, $CR$, and RHS terms, $RHS$ :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import flopy\n",
    "from os import system\n",
    "\n",
    "nrow, ncol = 1, 10000\n",
    "M0 = flopy.modflow.Modflow(modelname='model', exe_name='../mf2005.exe')\n",
    "flopy.modflow.ModflowDis(M0, nlay=1, nrow=nrow, ncol=ncol, nper=1, delr=1., delc=1., top=0., botm=-b, steady=True, \n",
    "                         perlen=1., nstp=1)\n",
    "flopy.modflow.ModflowBas(M0, ibound=np.hstack([np.ones([nrow, ncol-1], dtype=int), -1*np.ones([1,1])]), \n",
    "                         strt=BC1h*np.ones([nrow, ncol], dtype=float))\n",
    "flopy.modflow.ModflowLpf(M0, hk=K, vka=-999., ss=-999., sy=-999., ipakcb=53)\n",
    "flopy.modflow.ModflowRch(M0, nrchop=1, rech=R, ipakcb=53)\n",
    "flopy.modflow.ModflowPcg(M0, hclose=1e-6, rclose=1e-6)\n",
    "flopy.modflow.ModflowOc(M0, stress_period_data={(0,0): ['save head', 'save budget']})\n",
    "M0.write_input()\n",
    "#success, buff = M0.run_model(silent=True)\n",
    "system('mf2005_stream.exe < model.in')\n",
    "h = np.atleast_2d(flopy.utils.binaryfile.HeadFile('model.hds').get_data()[0,0,:])\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "plt.subplots(figsize=[8,2])\n",
    "#plt.plot(X/1000., h[0, range(500, np.shape(h)[1], 1000)], 'ko--', mfc='none', label='forward solution, numerical')\n",
    "plt.plot(X/1000., np.ravel(h), 'k--', mfc='none', label='forward solution, numerical')\n",
    "plt.xlabel('$x$ (km)')\n",
    "plt.ylabel('$h$ (m)')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Import RHS vector:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(1, 10000)\n"
     ]
    }
   ],
   "source": [
    "RHS = np.atleast_2d(np.loadtxt('RHS.arr').flatten())\n",
    "print(np.shape(RHS))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assemble A matrix from CR vector terms:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "'''CR = np.reshape(np.loadtxt('CR.arr').flatten(), [1,ncol])\n",
    "CR = np.vstack([np.zeros(ncol),       CR, np.zeros(ncol)])\n",
    "CR = np.hstack([np.zeros([nrow+2,1]), CR, np.zeros([nrow+2,1])])\n",
    "A = np.zeros([nrow*ncol, nrow*ncol])\n",
    "i = 0\n",
    "for r in range(1, nrow+1):\n",
    "    for c in range(1, ncol+1):\n",
    "        if i-1 > -1:\n",
    "            A[i,i-1] = CR[r,c-1]\n",
    "        A[i,i] = -CR[r,c-1]-CR[r,c]\n",
    "        if i+1 < nrow*ncol-1:\n",
    "            A[i,i+1] = CR[r,c]\n",
    "        i +=1\n",
    "A[-1,-1] = 0.\n",
    "dAdK = A/K\n",
    "print(dAdK)\n",
    "A = None''';"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assemble A matrix analytically:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-10.  10.   0. ...   0.   0.   0.]\n",
      " [ 10. -20.  10. ...   0.   0.   0.]\n",
      " [  0.  10. -20. ...   0.   0.   0.]\n",
      " ...\n",
      " [  0.   0.   0. ... -20.  10.   0.]\n",
      " [  0.   0.   0. ...  10. -20.  10.]\n",
      " [  0.   0.   0. ...   0.  10. -20.]]\n"
     ]
    }
   ],
   "source": [
    "from scipy.sparse import dia_matrix\n",
    "n = int(L)\n",
    "ex = np.ones(n)\n",
    "data = np.array([ex*b, ex*-2.*b, ex*b])\n",
    "offsets = np.array([-1, 0, 1])\n",
    "dAdK = dia_matrix((data, offsets), shape=(n, n)).toarray()\n",
    "dAdK[0,0] = -b\n",
    "#dAdK_test[:,-1] = 0.\n",
    "#dAdK_test[-1,:] = 0.\n",
    "print(dAdK)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(1, 10000) (1, 10000) (10000, 10000)\n"
     ]
    }
   ],
   "source": [
    "print(psi.shape, h.shape, dAdK.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
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      "text/plain": [
       "<IPython.core.display.HTML object>"
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     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.749250 (-3.750000)\n"
     ]
    }
   ],
   "source": [
    "set_background('rgba(0, 200, 0, 0.2)')\n",
    "print('%.6f (%.6f)'% (np.float(-psi@dAdK@h.T), benchmark))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<img src onerror=\"var cell = this.closest('.code_cell');var editor = cell.querySelector('.input_area');editor.style.background='rgba(200, 0, 0, 0.2)';this.parentNode.removeChild(this)\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " \n",
      "!!! IGNORE WORKINGS BELOW !!!\n",
      " \n"
     ]
    }
   ],
   "source": [
    "set_background('rgba(200, 0, 0, 0.2)')\n",
    "print(' \\n!!! IGNORE WORKINGS BELOW !!!\\n ')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assemble A matrix and RHS vector:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "CR = np.reshape(np.loadtxt('CR.arr').flatten(), [1,ncol])\n",
    "CR = np.vstack([np.zeros(ncol),       CR, np.zeros(ncol)])\n",
    "CR = np.hstack([np.zeros([nrow+2,1]), CR, np.zeros([nrow+2,1])])\n",
    "CC  = np.zeros(np.shape(CR))\n",
    "A = np.zeros([nrow*ncol, nrow*ncol])\n",
    "i = 0\n",
    "for r in range(1, nrow+1):\n",
    "    for c in range(1, ncol+1):\n",
    "        if r==nrow and c==ncol:\n",
    "            continue\n",
    "        if i-ocol > -1:\n",
    "            A[i, i-ocol] = CC[r-1,c]\n",
    "        if i-1 > -1:\n",
    "            A[i, i-1] = CR[r,c-1]\n",
    "        A[i, i] = -CC[r-1,c]-CR[r,c-1]-CC[r,c]-CR[r,c]\n",
    "        if i+1 < nrow*ncol-1:\n",
    "            A[i, i+1] = CR[r,c]\n",
    "        if i+ocol < nrow*ncol-1:\n",
    "            A[i, i+ocol] = CC[r,c]\n",
    "        i +=1\n",
    "#A = A[:-1,:-1]\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "RHS = np.atleast_2d(np.loadtxt('RHS.arr'))\n",
    "print(np.shape(RHS))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "M0 = flopy.modflow.Modflow(modelname='model', exe_name='../mf2005.exe')\n",
    "flopy.modflow.ModflowDis(M0, nlay=1, nrow=nrow, ncol=ncol, nper=1, delr=1., delc=1., top=0., botm=-b, steady=True, \n",
    "                         perlen=1., nstp=1)\n",
    "flopy.modflow.ModflowBas(M0, ibound=np.hstack([np.ones([nrow, ncol-1], dtype=int), -1*np.ones([1,1])]), \n",
    "                         strt=BC1h*np.ones([nrow, ncol], dtype=float))\n",
    "flopy.modflow.ModflowLpf(M0, hk=K, vka=-999., ss=-999., sy=-999., ipakcb=53)\n",
    "flopy.modflow.ModflowRch(M0, nrchop=1, rech=R, ipakcb=53)\n",
    "flopy.modflow.ModflowPcg(M0, hclose=1e-6, rclose=1e-6)\n",
    "flopy.modflow.ModflowOc(M0, stress_period_data={(0,0): ['save head', 'save budget']})\n",
    "M0.write_input()\n",
    "#success, buff = M0.run_model(silent=True)\n",
    "system('mf2005_stream.exe < model.in')\n",
    "\n",
    "h = np.reshape(flopy.utils.binaryfile.HeadFile('model.hds').get_data()[0,0,:], [1,ncol]).T\n",
    "CR = np.reshape(np.loadtxt('CR.arr').flatten(), [1,ncol])\n",
    "CR = np.vstack([np.zeros(ncol),       CR, np.zeros(ncol)])\n",
    "CR = np.hstack([np.zeros([nrow+2,1]), CR, np.zeros([nrow+2,1])])\n",
    "CC  = np.zeros(np.shape(CR))\n",
    "A = np.zeros([nrow*ncol, nrow*ncol])\n",
    "i = 0\n",
    "for r in range(1, nrow+1):\n",
    "    for c in range(1, ncol+1):\n",
    "        if i-1 > -1:\n",
    "            A[i,i-1] = CR[r,c-1]\n",
    "        A[i,i] = -CR[r,c-1]-CR[r,c]\n",
    "        if i+1 < nrow*ncol-1:\n",
    "            A[i,i+1] = CR[r,c]\n",
    "        i +=1\n",
    "A[-1,-1] = 0.\n",
    "A0 = A\n",
    "print(A0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.sparse import dia_matrix\n",
    "n = 10000\n",
    "ex = np.ones(n)\n",
    "data = np.array([100*ex, -200*ex, 100*ex])\n",
    "offsets = np.array([-1, 0, 1])\n",
    "test = dia_matrix((data, offsets), shape=(n, n)).toarray()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dpar = 1e-4\n",
    "M0 = flopy.modflow.Modflow(modelname='model', exe_name='../mf2005.exe')\n",
    "flopy.modflow.ModflowDis(M0, nlay=1, nrow=nrow, ncol=ncol, nper=1, delr=1., delc=1., top=0., botm=-b, steady=True, \n",
    "                         perlen=1., nstp=1)\n",
    "flopy.modflow.ModflowBas(M0, ibound=np.hstack([np.ones([nrow, ncol-1], dtype=int), -1*np.ones([1,1])]), \n",
    "                         strt=BC1h*np.ones([nrow, ncol], dtype=float))\n",
    "flopy.modflow.ModflowLpf(M0, hk=K+K*dpar, vka=-999., ss=-999., sy=-999., ipakcb=53)\n",
    "flopy.modflow.ModflowRch(M0, nrchop=1, rech=R, ipakcb=53)\n",
    "flopy.modflow.ModflowPcg(M0, hclose=1e-6, rclose=1e-6)\n",
    "flopy.modflow.ModflowOc(M0, stress_period_data={(0,0): ['save head', 'save budget']})\n",
    "M0.write_input()\n",
    "#success, buff = M0.run_model(silent=True)\n",
    "system('mf2005_stream.exe < model.in')\n",
    "\n",
    "h = np.reshape(flopy.utils.binaryfile.HeadFile('model.hds').get_data()[0,0,:], [1,ncol]).T\n",
    "CR = np.reshape(np.loadtxt('CR.arr').flatten(), [1,ncol])\n",
    "CR = np.vstack([np.zeros(ncol),       CR, np.zeros(ncol)])\n",
    "CR = np.hstack([np.zeros([nrow+2,1]), CR, np.zeros([nrow+2,1])])\n",
    "CC  = np.zeros(np.shape(CR))\n",
    "A = np.zeros([nrow*ncol, nrow*ncol])\n",
    "i = 0\n",
    "for r in range(1, nrow+1):\n",
    "    for c in range(1, ncol+1):\n",
    "        if i-1 > -1:\n",
    "            A[i,i-1] = CR[r,c-1]\n",
    "        A[i,i] = -CR[r,c-1]-CR[r,c]\n",
    "        if i+1 < nrow*ncol-1:\n",
    "            A[i,i+1] = CR[r,c]\n",
    "        i +=1\n",
    "A1 = A\n",
    "print(A1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dAdK = (A1-A0)/(K*dpar)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "dAdK"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "psi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dAdK"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "h"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.float(-psi@dAdK@h)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.shape(psi), np.shape(dAdK), np.shape(h)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "h_analyt = np.atleast_2d([R/K/2./b*(L**2.-x**2.) for x in X]).T\n",
    "dhdK_analyt = np.atleast_2d([R/K**2./2./b*(x**2.-L**2.) for x in [5000.]])\n",
    "def a(x, xp, K, b, L):\n",
    "    if x>xp:\n",
    "        a = L-x\n",
    "    else:\n",
    "        a = L-xp\n",
    "    return a/K/b\n",
    "psi_analyt = np.atleast_2d([a(x, ocol, K, b, L) for x in X])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "psi_analyt_inv = np.linalg.pinv(psi_analyt)\n",
    "h_analyt_inv = np.linalg.pinv(h_analyt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.shape(psi_analyt_inv)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dAdK_analyt = -psi_analyt_inv@dhdK_analyt@h_analyt_inv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.float(-psi_analyt@dAdK_analyt@h_analyt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.shape(dAdK_analyt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.float(-psi@test@h)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "psi.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "h.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "test.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def min_test(pars):\n",
    "    print(pars)\n",
    "    global psi, h\n",
    "    n = 10000\n",
    "    ex = np.ones(n)\n",
    "    data = np.array([ex*pars[0], ex*pars[1], ex*pars[2]])\n",
    "    offsets = np.array([-1, 0, 1])\n",
    "    dhdA_test = dia_matrix((data, offsets), shape=(n, n)).toarray()\n",
    "    return -3.75-np.float(-psi@dhdA_test@h)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.optimize import least_squares\n",
    "pars0 = np.array([1., 1., 1.])\n",
    "res = least_squares(min_test, pars0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 10000\n",
    "ex = np.ones(n)\n",
    "data = np.array([ex*res.x[0], ex*res.x[1], ex*res.x[2]])\n",
    "offsets = np.array([-1, 0, 1])\n",
    "dhdA_opt = dia_matrix((data, offsets), shape=(n, n)).toarray()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dhdA_opt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "-psi@dhdA_opt@h"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "res.x*K*dpar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "CR = np.atleast_2d(np.loadtxt('CR.arr').flatten())\n",
    "CR = np.vstack([np.zeros(ncol),       CR, np.zeros(ncol)])\n",
    "CR = np.hstack([np.zeros([nrow+2,1]), CR, np.zeros([nrow+2,1])])\n",
    "np.shape(CR)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.unique(RHS)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "h.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def min_test(pars):\n",
    "    print(pars)\n",
    "    global psi, h\n",
    "    n = 10000\n",
    "    ex = np.ones(n)\n",
    "    data = np.array([ex*pars, ex*-2.*pars, ex*pars])\n",
    "    offsets = np.array([-1, 0, 1])\n",
    "    dhdA_test = dia_matrix((data, offsets), shape=(n, n)).toarray()\n",
    "    dhdA_test[0,0] = -pars\n",
    "    dhdA_test[:,-1] = 0.\n",
    "    dhdA_test[-1,:] = 0.\n",
    "    return -3.75-np.float(-psi@dhdA_test@h)\n",
    "\n",
    "from scipy.optimize import least_squares\n",
    "pars0 = 1.\n",
    "res = least_squares(min_test, pars0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dpar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "h"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "RHS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "psi.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "h.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "A.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dAdK.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dAdK"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 10000\n",
    "ex = np.ones(n)\n",
    "data = np.array([ex*b, ex*-2.*b, ex*b])\n",
    "offsets = np.array([-1, 0, 1])\n",
    "dhdA_test = dia_matrix((data, offsets), shape=(n, n)).toarray()\n",
    "dhdA_test[0,0] = -b\n",
    "#dhdA_test[:,-1] = 0.\n",
    "#dhdA_test[-1,:] = 0.\n",
    "np.float(-psi@dhdA_test@h)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dhdA_test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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