PDE example.ipynb

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   "source": [
    "########################################################\n",
    "### This file is used to generate Table 7-8, Fig 4-5 ###\n",
    "########################################################\n",
    "\n",
    "- [Forward Problem](#Forward-Problem)\n",
    "    - [Verify Assumption 1](#Verify-Assumption-1)\n",
    "        - [Table 7](#Table-7)\n",
    "        - [Table 8](#Table-8)\n",
    "    - [Verify Lemma 1](#Verify-Lemma-1)\n",
    "        - [Left plot in Figure 4](#Left-plot-in-Figure-4)\n",
    "    - [Verify Theorem 3.1](#Verify-Theorem-3.1)\n",
    "        - [Right plot in Figure 4](#Right-plot-in-Figure-4)\n",
    "- [Inverse Problem](#Inverse-Problem)\n",
    "    - [Verify Assumption 2](#Verify-Assumption-2)\n",
    "    - [Verify Theorem 4.2](#Verify-Theorem-4.2)\n",
    "        - [Figure 5](#Figure-5)"
   ]
  },
  {
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    "ExecuteTime": {
     "end_time": "2019-11-22T16:56:19.339829Z",
     "start_time": "2019-11-22T16:56:19.335388Z"
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   "source": [
    "<font color=red>**Note(Important!!!!!)** \n",
    "Since this file would require data from Data directory, you will need to generate the data files, there are two ways to generate the data.</font>\n",
    "\n",
    "- Way 1: run 'GenerateData.ipynb' or 'GenerateData.py' to get QoI values at quadrature points in order to get coefficients in PCE; and all the ratio evaluations\n",
    "- Way 2: run 'GenerateData_ParallelVersion.ipynb' and set multiple processors"
   ]
  },
  {
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     "end_time": "2021-07-01T03:19:32.914969Z",
     "start_time": "2021-07-01T03:19:30.623582Z"
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   },
   "outputs": [],
   "source": [
    "import os\n",
    "import scipy.io as sio #for the i/o\n",
    "import numpy as np\n",
    "import numpy.polynomial.hermite_e as H \n",
    "from math import factorial\n",
    "from scipy.stats import norm\n",
    "from scipy.stats import gaussian_kde as kde\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "from matplotlib import pyplot as plt\n",
    "\n",
    "import ipywidgets as widgets\n",
    "from ipywidgets import interact, interactive, fixed, interact_manual\n",
    "from matplotlib import cm\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:19:32.944373Z",
     "start_time": "2021-07-01T03:19:32.919054Z"
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   "source": [
    "####### Plot Formatting ######\n",
    "plt.rc('lines', linewidth = 1.5)\n",
    "plt.rc('xtick', labelsize = 14)\n",
    "plt.rc('ytick', labelsize = 14)\n",
    "plt.rc('legend',fontsize=14)\n",
    "# plt.rcParams[\"font.family\"] = \"serif\"\n",
    "plt.rcParams['axes.labelsize'] = 20\n",
    "plt.rcParams['axes.titlesize'] = 12\n",
    "plt.rcParams['lines.markersize'] = 6\n",
    "plt.rcParams['figure.figsize'] = (8.0, 6.0)"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Problem:\n",
    "\n",
    "\\begin{align*}\n",
    "- \\nabla\\cdot(A\\nabla u) &= (e^{\\lambda_1}\\lambda_1^2\\pi^2 + e^{\\lambda_2}\\lambda_2^2\\pi^2)u \\\\\n",
    "u &= 0 \\, \\text{ on } \\Gamma_0 \\,\\text{(   Left edge)}\\\\\n",
    "(A\\nabla u)\\cdot n &= -e^{\\lambda_2}\\lambda_2\\pi \\sin\\lambda_1\\pi x\\sin \\lambda_2\\pi y \\, \\text{ on } \\Gamma_1  \\, \\text{(   Top edge)}\\\\\n",
    "(A\\nabla u)\\cdot n &= e^{\\lambda_2}\\lambda_2\\pi \\sin\\lambda_1\\pi x\\sin \\lambda_2\\pi y \\, \\text{ on } \\Gamma_2  \\,\\text{(   Bottom edge)}\\\\\n",
    "(A\\nabla u)\\cdot n &= e^{\\lambda_1}\\lambda_1\\pi \\cos\\lambda_1\\pi x\\cos \\lambda_2\\pi y \\, \\text{ on } \\Gamma_3  \\,\\text{(   Right edge)}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "where \n",
    "$$ A = \\begin{bmatrix} e^{\\lambda_1} & 0 \\\\ 0 & e^{\\lambda_2} \\end{bmatrix} $$\n",
    "and $\\Omega = [0,1]\\times [0,1]$.\n",
    "\n",
    "<font color = red>**Exact solution:**\n",
    "$$ u = \\sin \\lambda_1\\pi x \\cos \\lambda_2 \\pi y$$\n",
    "</font>\n",
    "\n",
    "QoI is:\n",
    " \n",
    "\\begin{align*}\n",
    "Q(\\lambda_1,\\lambda_2) &=  \\frac{1}{(b-a)(d-c)}\\int_{c}^{d} \\int_{a}^{b} u(x,y,\\lambda_1,\\lambda_2)\\, dx\\, dy \\\\\n",
    "&= \\frac{1}{(b-a)(d-c)} \\int_{a}^{b} \\sin \\lambda_1\\pi x \\, dx \\int_{c}^{d} \\cos \\lambda_2 \\pi y \\, dy\\\\\n",
    "&= \\frac{1}{(b-a)(d-c)} \\left( \\frac{\\cos \\lambda_1\\pi a - \\cos\\lambda_1\\pi b}{\\lambda_1\\pi}\\right) \\left( \\frac{\\sin \\lambda_2\\pi d - \\sin \\lambda_2\\pi c}{\\lambda_2\\pi}\\right) \\\\\n",
    "&= \\frac{(\\cos \\lambda_1\\pi a - \\cos\\lambda_1\\pi b)(\\sin \\lambda_2\\pi d - \\sin\\lambda_2\\pi c) }{(b-a)(d-c)\\lambda_1\\lambda_2\\pi^2} \n",
    "\\end{align*}\n"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### When standard normal\n",
    "</font >\n",
    "\n",
    "$$ \\Phi_{ij}(x,y) = \\Phi_i(x) \\Phi_j(y)$$\n",
    "\n",
    "$$ (\\Phi_{ij}, \\Phi_{kl}) = \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} \\Phi_{ij}(x,y)\\Phi_{kl}(x,y) e^{-\\frac{x^2+y^2}{2}}\\, dx\\, dy = (\\sqrt{2\\pi})^2 i! j! \\delta_{ik}\\delta_{jl}$$\n",
    "\n",
    "\n",
    "$$ Q(\\lambda_1, \\lambda_2) = \\sum_{i,j=0}^{\\infty} q_{ij} \\Phi_{ij}(\\lambda_1,\\lambda_2)$$\n",
    "\n",
    "$$ q_{ij} = \\frac{1}{(\\Phi_{ij},\\Phi_{ij})} \\biggr(Q(\\lambda_1, \\lambda_2), \\  \\Phi_{ij}(\\lambda_1,\\lambda_2)\\biggr) = \\frac{1}{(\\sqrt{2\\pi})^2 i! j! } \\biggr(Q(\\lambda_1, \\lambda_2), \\  \\Phi_{ij}(\\lambda_1,\\lambda_2)\\biggr)$$\n",
    "\n",
    "**Gauss-Hermite Quadrature**\n",
    " \n",
    "\\begin{align*}\n",
    "\\biggr(Q(\\lambda_1, \\lambda_2), \\  \\Phi_{ij}(\\lambda_1,\\lambda_2)\\biggr) &= \\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty} Q(\\lambda_1,\\lambda_2) \\Phi_i(\\lambda_1) e^{-\\frac{\\lambda_1^2}{2}} \\, d\\lambda_1 \\Phi_j(\\lambda_2) e^{-\\frac{\\lambda_2^2}{2}} \\, d\\lambda_2\\\\\n",
    "&= \\int_{-\\infty}^{\\infty} \\biggr( \\sum_{k=1}^n w_k Q(\\lambda_1^{(k)}, \\lambda_2)\\Phi_i(\\lambda_1^{(k)}) \\biggr) \\Phi_j(\\lambda_2) e^{-\\frac{\\lambda_2^2}{2}} \\, d\\lambda_2\\\\ \n",
    "&= \\sum_{l=1}^n  w_l \\biggr( \\sum_{k=1}^n w_k Q(\\lambda_1^{(k)}, \\lambda_2^{(l)})\\Phi_i(\\lambda_1^{(k)}) \\biggr) \\Phi_j(\\lambda_2^{(l)})\n",
    "\\end{align*}"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### When generalized normal\n",
    "\n",
    "**Quadrature Rule**\n",
    "\n",
    "<font color = red>\n",
    "If \n",
    "$$ \\int_{-\\infty}^{\\infty} f(x) e^{-\\frac{x^2}{2}}\\, dx \\approx \\sum_{i=1}^n w_if(x_i)$$\n",
    "we will have\n",
    "$$ \\int_{-\\infty}^{\\infty} f(x) e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}\\, dx \\approx \\sigma \\sum_{i=1}^n w_if(\\mu + \\sigma x_i)$$\n",
    "</font >\n",
    "\n",
    "If $\\lambda_1$, $\\lambda_2$ don't follow standard normal $N(0,1)$, assume $\\lambda_1\\sim N(\\mu_1, \\sigma_1^2)$, $\\lambda_2\\sim N(\\mu_2, \\sigma_2^2)$, we will have \n",
    "\n",
    "$$ Q(\\lambda_1, \\lambda_2) = \\sum_{i,j=0}^{\\infty} q_{ij} \\Phi_{ij}\\left(\\frac{\\lambda_1 - \\mu_1}{\\sigma_1},\\frac{\\lambda_2 - \\mu_2}{\\sigma_2}\\right)$$\n",
    "\n",
    "\n",
    "\n",
    "Still use above $\\Phi_{ij}$, but in order to make it orthogonal, we should have basis like\n",
    "\n",
    "$$ \\Phi_{ij}\\left(\\frac{\\lambda_1 - \\mu_1}{\\sigma_1},\\frac{\\lambda_2 - \\mu_2}{\\sigma_2}\\right) $$\n",
    "\n",
    "then \n",
    "\n",
    "\\begin{align*}\n",
    "&\\,\\,\\, \\left(  \\Phi_{ij}\\left(\\frac{\\lambda_1 - \\mu_1}{\\sigma_1},\\frac{\\lambda_2 - \\mu_2}{\\sigma_2}\\right),  \\Phi_{kl}\\left(\\frac{\\lambda_1 - \\mu_1}{\\sigma_1},\\frac{\\lambda_2 - \\mu_2}{\\sigma_2}\\right)  \\right)\\\\\n",
    "&= \\left(  \\Phi_{i}\\left(\\frac{\\lambda_1 - \\mu_1}{\\sigma_1}\\right)\\Phi_{j}\\left(\\frac{\\lambda_2 - \\mu_2}{\\sigma_2}\\right),  \\Phi_{k}\\left(\\frac{\\lambda_1 - \\mu_1}{\\sigma_1}\\right) \\Phi_l\\left(\\frac{\\lambda_2 - \\mu_2}{\\sigma_2}\\right)  \\right)\\\\\n",
    "&= \\int_{R} \\int_{R} \\Phi_{i}\\left(\\frac{\\lambda_1 - \\mu_1}{\\sigma_1}\\right)\\Phi_{j}\\left(\\frac{\\lambda_2 - \\mu_2}{\\sigma_2}\\right)  \\Phi_{k}\\left(\\frac{\\lambda_1 - \\mu_1}{\\sigma_1}\\right) \\Phi_l\\left(\\frac{\\lambda_2 - \\mu_2}{\\sigma_2}\\right)e^{-\\frac{(\\lambda_1-\\mu_1)^2}{2\\sigma_1^2}} e^{-\\frac{(\\lambda_2-\\mu_2)^2}{2\\sigma_2^2}} d\\lambda_1 d\\lambda_2 \\\\\n",
    "&= \\sigma_1\\sigma_2 \\int_{R} \\int_{R} \\Phi_{i}(y_1)\\Phi_j(y_2)\\Phi_{k}(y_1)\\Phi_l(y_2) e^{-\\frac{y_1^2}{2}}e^{-\\frac{y_2^2}{2}}\\, dy_1\\, dy_2 \\\\\n",
    "&= \\sigma_1\\sigma_2 2\\pi i! j! \\delta_{ik}\\delta_{jl}\n",
    "\\end{align*}\n",
    "\n",
    "\n",
    "Since \n",
    "$$ Q(\\lambda_1, \\lambda_2) = \\sum_{i,j=0}^{\\infty} q_{ij} \\Phi_{ij}\\left(\\frac{\\lambda_1 - \\mu_1}{\\sigma_1},\\frac{\\lambda_2 - \\mu_2}{\\sigma_2}\\right)$$\n",
    "then\n",
    " \n",
    "\\begin{align*}\n",
    "q_{ij} &= \\frac{1}{\\left(\\Phi_{ij}\\left(\\frac{\\lambda_1 - \\mu_1}{\\sigma_1},\\frac{\\lambda_2 - \\mu_2}{\\sigma_2}\\right),\\Phi_{ij}\\left(\\frac{\\lambda_1 - \\mu_1}{\\sigma_1},\\frac{\\lambda_2 - \\mu_2}{\\sigma_2}\\right)\\right)} \\biggr(Q(\\lambda_1, \\lambda_2), \\  \\Phi_{ij}\\left(\\frac{\\lambda_1 - \\mu_1}{\\sigma_1},\\frac{\\lambda_2 - \\mu_2}{\\sigma_2}\\right)\\biggr) \\\\\n",
    "&= \\frac{1}{\\sigma_1 \\sigma_2 2\\pi i! j! } \\biggr(Q(\\lambda_1, \\lambda_2), \\  \\Phi_{ij}\\left(\\frac{\\lambda_1 - \\mu_1}{\\sigma_1},\\frac{\\lambda_2 - \\mu_2}{\\sigma_2}\\right)\\biggr)\\\\\n",
    "&= \\frac{1}{\\sigma_1 \\sigma_2 2\\pi i! j! } \\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty} Q(\\lambda_1,\\lambda_2) \\Phi_i\\left(\\frac{\\lambda_1-\\mu_1}{\\sigma_1}\\right) e^{-\\frac{(\\lambda_1-\\mu_1)^2}{2\\sigma_1^2}} \\, d\\lambda_1 \\Phi_j\\left(\\frac{\\lambda_2-\\mu_2}{\\sigma_2}\\right) e^{-\\frac{(\\lambda_2-\\mu_2)^2}{2\\sigma_2^2}} \\, d\\lambda_2\\\\\n",
    "&= \\frac{1}{\\sigma_1 \\sigma_2 2\\pi i! j! } \\int_{-\\infty}^{\\infty} \\sigma_1 \\biggr( \\sum_{k=1}^n w_k Q(\\mu_1+\\sigma_1\\lambda_1^{(k)}, \\lambda_2)\\Phi_i(\\lambda_1^{(k)}) \\biggr) \\Phi_j\\left(\\frac{\\lambda_2 - \\mu_2}{\\sigma_2}\\right) e^{-\\frac{(\\lambda_2-\\mu_2)^2}{2\\sigma_2^2}} \\, d\\lambda_2\\\\ \n",
    "&= \\frac{1}{\\sigma_1 \\sigma_2 2\\pi i! j! } \\sigma_2 \\sum_{l=1}^n  w_l \\sigma_1 \\biggr( \\sum_{k=1}^n w_k Q(\\mu_1+\\sigma_1\\lambda_1^{(k)}, \\mu_2+\\sigma_2\\lambda_2^{(l)})\\Phi_i(\\lambda_1^{(k)}) \\biggr) \\Phi_j(\\lambda_2^{(l)})\\\\\n",
    "&= \\frac{1}{2\\pi i! j! } \\sum_{l=1}^n  w_l \\biggr( \\sum_{k=1}^n w_k Q(\\mu_1+\\sigma_1\\lambda_1^{(k)}, \\mu_2+\\sigma_2\\lambda_2^{(l)})\\Phi_i(\\lambda_1^{(k)}) \\biggr) \\Phi_j(\\lambda_2^{(l)})\n",
    "\\end{align*}"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assume \n",
    "$$ \\lambda_1 \\sim N(\\mu_1, \\sigma_1^2) = N(0, 0.1^2) \\ \\ \\ \\lambda_2 \\sim N(\\mu_2, \\sigma_2^2) = N(0, 0.1^2) $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:19:32.957193Z",
     "start_time": "2021-07-01T03:19:32.952194Z"
    }
   },
   "outputs": [],
   "source": [
    "proc_size = 25   # this number is determined by the number of data files"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:19:32.970411Z",
     "start_time": "2021-07-01T03:19:32.962788Z"
    }
   },
   "outputs": [],
   "source": [
    "mu1 = 0\n",
    "mu2 = 0\n",
    "sigma1 = 0.1\n",
    "sigma2 = 0.1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Get $Q_n(\\lambda_1,\\lambda_2)$ (need to compute coef $q_{ij}$)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:19:34.009575Z",
     "start_time": "2021-07-01T03:19:32.974797Z"
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   "source": [
    "def Hermite_2d(i,j,x,y):\n",
    "    '''\n",
    "    Phi_{i,j}(x,y) = Phi_i(x) * Phi_j(y)  (left: 2d; right: 1d)\n",
    "    '''\n",
    "    c = np.zeros((20,20))\n",
    "    c[i,j] = 1\n",
    "    return H.hermeval2d(x, y, c)\n",
    " \n",
    "Q_FEM_quad = np.zeros(int(400))   #already include information of mu1, mu2, sigma1, sigma2\n",
    "for i in range(proc_size):          \n",
    "    filename = os.path.join(os.getcwd(), \"Data\", \"Q_FEM_quad_\") + str(i) + \".mat\"\n",
    "    partial_data = sio.loadmat(filename)\n",
    "    Q_FEM_quad += partial_data['Q_FEM'].reshape(int(400))\n",
    "\n",
    "def Phi(n):\n",
    "    #define H_n\n",
    "    coeffs = [0]*(n+1)\n",
    "    coeffs[n] = 1\n",
    "    return coeffs\n",
    "\n",
    "def q(i,j):\n",
    "    '''\n",
    "    copmute coefficient q_{ij}    \n",
    "    Set up Gauss-Hermite quadrature, weighting function is exp^{-x^2}\n",
    "    '''\n",
    "    x, w=H.hermegauss(20)     \n",
    "    Q=sum([w[ldx]*sum([w[kdx] * Q_FEM_quad[ldx*20+kdx] * H.hermeval(x[kdx],Phi(i)) for kdx in range(20)])*H.hermeval(x[ldx],Phi(j)) for ldx in range(20)])     \n",
    "    q= Q/(2*np.pi*factorial(i)*factorial(j))\n",
    "    \n",
    "    return q\n",
    "\n",
    "qij = np.zeros((10,10))\n",
    "for i in range(10):\n",
    "    for j in range(10):\n",
    "        qij[i,j] = q(i,j)\n",
    "        \n",
    "def Q(n,x,y):\n",
    "    result = 0 \n",
    "    for i in range(n+1):\n",
    "        for j in range(n+1):\n",
    "            if i+j <=n:\n",
    "                result += qij[i,j]*Hermite_2d(i,j,(x-mu1)/sigma1,(y-mu2)/sigma2)\n",
    "    return result\n",
    "\n",
    "def Qexact(x,y,a=0.4,b=0.6,c=0.4,d=0.6):\n",
    "    sol = (np.cos(x*np.pi*a)-np.cos(x*np.pi*b))*(np.sin(y*np.pi*d)-np.sin(y*np.pi*c))/((b-a)*(d-c)*x*y*np.pi**2)\n",
    "    return sol"
   ]
  },
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   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:19:34.766482Z",
     "start_time": "2021-07-01T03:19:34.012986Z"
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   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 576x504 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Visualize the error between PCE and exact\n",
    "fig = plt.figure(figsize=(8,7))\n",
    "ax = fig.gca(projection='3d')\n",
    "\n",
    "# Make data.\n",
    "X = np.arange(-0.5, 0.5, 0.1)\n",
    "Y = np.arange(-0.5, 0.5, 0.1)\n",
    "X, Y = np.meshgrid(X, Y)\n",
    "# Z = Q(1, X, Y)\n",
    "# Z = Q(5,X,Y) - Qexact(X,Y)\n",
    "Z = Qexact(X,Y)\n",
    "\n",
    "# Plot the surface.\n",
    "surf = ax.plot_surface(X, Y, Z, cmap=cm.coolwarm,\n",
    "                       linewidth=0, antialiased=False)\n",
    "\n",
    "# Customize the z axis.\n",
    "ax.set_zlim(-0.5, 0.5)        #Qexact zlim(-0.5,0.5); Qi-Q_exact zlim(-1.2,1.2)\n",
    "ax.set_xlabel(\"$\\lambda_1$\", fontsize = 16)\n",
    "ax.set_ylabel(\"$\\lambda_2$\", fontsize = 16)\n",
    "ax.tick_params(direction='out', length=6, width=2, pad=1 ,colors='black',  \\\n",
    "               grid_color='r', grid_alpha=0.5, labelsize=14)    #pad:dist b/t label and axis\n",
    "cbaxes = fig.add_axes([0.88, 0.25, 0.03, 0.5])     #1st:horizontal, 2&4th:vertical, 3rd:width\n",
    "fig.colorbar(surf, shrink=5, aspect=14, cax=cbaxes)\n",
    "plt.show()\n",
    "# fig.savefig(\"images/2Q_exact_5std\")   #2Q_exact_5std, 2Q1_5std"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Forward Problem\n",
    "Assume \n",
    "$$ \\lambda_1 \\sim N(\\mu_1, \\sigma_1^2) = N(0, 0.1^2) \\ \\ \\ \\lambda_2 \\sim N(\\mu_2, \\sigma_2^2) = N(0, 0.1^2) $$\n",
    "\n",
    "### Verify Assumption 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:21:25.074114Z",
     "start_time": "2021-07-01T03:19:34.773394Z"
    }
   },
   "outputs": [],
   "source": [
    "##### Generate data in Table 4 and 5 #####\n",
    "def assumption1(n, J):\n",
    "    np.random.seed(123456)\n",
    "    lam1sample = np.random.normal(mu1, sigma1, J)\n",
    "    lam2sample = np.random.normal(mu2, sigma2, J)\n",
    "    pfprior_sample_n = Q(n, lam1sample, lam2sample)\n",
    "    pfprior_dens_n = kde(pfprior_sample_n)\n",
    "    x = np.linspace(-1, 1, 1000)\n",
    "    return np.round(np.max(np.abs(np.gradient(pfprior_dens_n(x), x))), 2), np.round(np.max(pfprior_dens_n(x)), 2)\n",
    "\n",
    "\n",
    "size_J = [int(1E3), int(1E4), int(1E5)]\n",
    "degree_n = [1, 2, 3, 4, 5]\n",
    "Bound_matrix, Lip_Bound_matrix = np.zeros((3, 5)), np.zeros((3, 5))\n",
    "for i in range(3):\n",
    "    for j in range(5):\n",
    "        n, J = degree_n[j], size_J[i]\n",
    "        Lip_Bound_matrix[i, j] = assumption1(n, J)[0]\n",
    "        Bound_matrix[i, j] = assumption1(n, J)[1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Table 7"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:21:25.091054Z",
     "start_time": "2021-07-01T03:21:25.081174Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Table 7\n",
      "Bound under certain n and J values\n",
      "[[2.63 2.63 2.61 2.61 2.61]\n",
      " [2.64 2.64 2.61 2.61 2.61]\n",
      " [2.6  2.6  2.57 2.57 2.57]]\n"
     ]
    }
   ],
   "source": [
    "###########################################\n",
    "################ Table 7 ##################\n",
    "###########################################\n",
    "print('Table 7')\n",
    "print('Bound under certain n and J values')\n",
    "print(Bound_matrix)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Table 8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:21:25.104560Z",
     "start_time": "2021-07-01T03:21:25.097565Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Table 8\n",
      "Lipschitz bound under certain n and J values\n",
      "[[12.11 12.11 11.95 11.95 11.95]\n",
      " [11.47 11.47 11.41 11.41 11.41]\n",
      " [11.07 11.07 10.86 10.86 10.86]]\n"
     ]
    }
   ],
   "source": [
    "###########################################\n",
    "################ Table 8 ##################\n",
    "###########################################\n",
    "print('Table 8')\n",
    "print('Lipschitz bound under certain n and J values')\n",
    "print(Lip_Bound_matrix)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:21:25.882853Z",
     "start_time": "2021-07-01T03:21:25.108985Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 576x432 with 0 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "d5fa49de411b4d0682b151d69afa9d20",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(IntSlider(value=1, description='n', max=5, min=1), IntSlider(value=1000, description='J'…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.plot_pushforward(n, J)>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#### Use plot to show the difference between the exact pushforward and approximate pushforward #####\n",
    "fig = plt.figure()\n",
    "def plot_pushforward(n,J):\n",
    "    #pfprior_dens = kde(Q_FEM)\n",
    "    \n",
    "    np.random.seed(123456)\n",
    "    lam1sample = np.random.normal(mu1,sigma1,J)\n",
    "    lam2sample = np.random.normal(mu2,sigma2,J)\n",
    "\n",
    "    pfprior_sample = Qexact(lam1sample,lam2sample)\n",
    "    pfprior_dens = kde(pfprior_sample)\n",
    "    \n",
    "    pfprior_sample_n = Q(n,lam1sample,lam2sample)\n",
    "    pfprior_dens_n = kde(pfprior_sample_n)\n",
    "    \n",
    "    fig.clear()\n",
    "    qplot = np.linspace(-1,1, num=1000)\n",
    "        \n",
    "    plt.plot(qplot,pfprior_dens(qplot),color='r', linestyle='-.', linewidth=4,label=\"$\\pi_{\\mathcal{D}}^{Q}$\")\n",
    "    plt.plot(qplot,pfprior_dens_n(qplot),label=\"$\\pi_{\\mathcal{D}}^{Q_n}$\")\n",
    "        \n",
    "    plt.title('Lipschitz const. = %4.2f and Bound = %2.2f' %(np.max(np.abs(np.gradient(pfprior_dens_n(qplot), qplot))),\n",
    "                                                                             np.max(pfprior_dens_n(qplot))))\n",
    "    plt.legend()\n",
    "    \n",
    "interact(plot_pushforward, \n",
    "         n = widgets.IntSlider(value=int(1),min=int(1),max=int(5),step=1),\n",
    "         J = widgets.IntSlider(value=int(1E3),min=int(1E3),max=int(1E5),step=int(1E3)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Verify Lemma 1\n",
    "**Print out Monte Carlo Approximation of $\t\\|\\pi_{\\mathcal{D}}^Q(q)-\\pi_{\\mathcal{D}}^{Q_n}(q)\\|_{L^r(\\mathcal{D_c})} $ where $r>0$ and $D_c=[-1,1]$**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:21:25.902269Z",
     "start_time": "2021-07-01T03:21:25.887177Z"
    }
   },
   "outputs": [],
   "source": [
    "##### Generate data for the left plot of Fig 3 #####\n",
    "# Define push-forward densities\n",
    "N_kde = int(1E4)\n",
    "N_mc = int(1E4)\n",
    "\n",
    "np.random.seed(123456)\n",
    "lam1sample = np.random.normal(mu1,sigma1,N_kde)\n",
    "lam2sample = np.random.normal(mu2,sigma2,N_kde)\n",
    "pfprior_dens = kde(Qexact(lam1sample,lam2sample))\n",
    "\n",
    "def pfprior_dens_n(n,x):\n",
    "    pfprior_sample_n =  Q(n,lam1sample,lam2sample)\n",
    "    pdf = kde(pfprior_sample_n)\n",
    "    return pdf(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:23:32.599714Z",
     "start_time": "2021-07-01T03:21:25.906009Z"
    }
   },
   "outputs": [],
   "source": [
    "# **Print out Monte Carlo Approximation of $\t\\|\\pi_{\\mathcal{D}}^Q(q)-\\pi_{\\mathcal{D}}^{Q_n}(q)\\|_{L^r(\\mathcal{D_c})} $ where $r>0$ and $D_c=[-1,1]$**\n",
    "np.random.seed(123456)\n",
    "qsample = np.random.uniform(-1,1,N_mc)\n",
    "\n",
    "def error_r_onD(r,n):\n",
    "    diff = (np.mean((np.abs(pfprior_dens_n(n,qsample) - pfprior_dens(qsample)))**r))**(1/r)\n",
    "    return diff\n",
    "\n",
    "error_r_D = np.zeros((5,5))\n",
    "for i in range(5):\n",
    "    for j in range(5):\n",
    "        error_r_D[i,j] = error_r_onD(i+1,j+1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:23:32.628053Z",
     "start_time": "2021-07-01T03:23:32.609100Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "L^r error on data space for Forward Problem\n",
      "\n",
      "[[5.78985998e-03 5.78986860e-03 3.66465835e-05 3.66458716e-05 4.68447031e-06]\n",
      " [1.00098540e-02 1.00098559e-02 6.48003418e-05 6.48018329e-05 6.83125067e-06]\n",
      " [1.29985177e-02 1.29985144e-02 8.36155928e-05 8.36174583e-05 7.98012310e-06]\n",
      " [1.51923261e-02 1.51923210e-02 9.70172096e-05 9.70187251e-05 8.72619888e-06]\n",
      " [1.68583067e-02 1.68583012e-02 1.07115873e-04 1.07116859e-04 9.26298356e-06]]\n"
     ]
    }
   ],
   "source": [
    "np.set_printoptions(linewidth=110) \n",
    "print('L^r error on data space for Forward Problem',end='\\n\\n')\n",
    "print(error_r_D) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:23:32.653535Z",
     "start_time": "2021-07-01T03:23:32.634719Z"
    }
   },
   "outputs": [],
   "source": [
    "#### To make it cleaner, create Directory \"images\" to store all the figures ####\n",
    "imagepath = os.path.join(os.getcwd(),\"images\")\n",
    "os.makedirs(imagepath,exist_ok=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Left plot in Figure 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:23:33.707431Z",
     "start_time": "2021-07-01T03:23:32.660388Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "###########################################\n",
    "######### The left plot of Fig 4 ##########\n",
    "###########################################\n",
    "fig = plt.figure()\n",
    "plt.xlim([0,6])\n",
    "marker = ['-D', '-o', '-v', '-s', '-.']\n",
    "for i in range(5):\n",
    "    plt.semilogy([1,2,3,4,5],error_r_D[i,:],marker[i],label='r = ' + np.str(i+1))    \n",
    "plt.xlabel('Order of PCE (n)')\n",
    "plt.ylabel('$L^r$'+' Error in Push-Forward on '+'$\\mathcal{D}$')\n",
    "plt.legend()\n",
    "# fig.savefig(\"images/2forward_error_D.png\")\n",
    "fig.savefig(\"images/Fig4(Left).png\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Verify Theorem 3.1\n",
    "**Print out Monte Carlo Approximation of $\t\\|\\pi_{\\mathcal{D}}^Q(Q(\\lambda))-\\pi_{\\mathcal{D}}^{Q_n}(Q_n(\\lambda))\\|_{L^2(\\Lambda)} $**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:23:57.149993Z",
     "start_time": "2021-07-01T03:23:33.714011Z"
    }
   },
   "outputs": [],
   "source": [
    "##### Generate data for the right plot of Fig 4 #####\n",
    "np.random.seed(123456)\n",
    "lam1_seed = np.random.normal(mu1,sigma1,int(1E4))\n",
    "lam2_seed = np.random.normal(mu2,sigma2,int(1E4))   #int(1E4) since Q_FEM size\n",
    "\n",
    "error_2_Lam = np.zeros(5)\n",
    "for i in range(5):\n",
    "    pfprior_sample = Qexact(lam1_seed,lam2_seed)\n",
    "    error_2_Lam[i] = (np.mean((np.abs(pfprior_dens_n(i+1,Q(i+1,lam1_seed,lam2_seed))\\\n",
    "                                      - pfprior_dens(pfprior_sample)))**2))**(1/2)   "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:23:57.232407Z",
     "start_time": "2021-07-01T03:23:57.154772Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "L^2 error on parameter space for Forward Problem\n",
      "\n",
      "[2.78836391e-02 2.78836376e-02 2.33315440e-04 2.33317057e-04 1.82851667e-05]\n"
     ]
    }
   ],
   "source": [
    "np.set_printoptions(linewidth=110)   \n",
    "print('L^2 error on parameter space for Forward Problem',end='\\n\\n')\n",
    "print(error_2_Lam)  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Right plot in Figure 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:23:58.007927Z",
     "start_time": "2021-07-01T03:23:57.236575Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "############################################\n",
    "######### The right plot of Fig 4 ##########\n",
    "############################################\n",
    "fig = plt.figure()\n",
    "plt.xlim([0,6])\n",
    "plt.semilogy([1,2,3,4,5], error_2_Lam, '-s' )#, label='$L^2(\\Lambda)$ error')\n",
    "plt.xlabel('Order of PCE (n)')\n",
    "plt.ylabel('$L^2$'+' Error in Push-Forward on '+'$\\Lambda$');\n",
    "# fig.savefig(\"images/2forward_error_lam.png\")\n",
    "fig.savefig(\"images/Fig4(Right).png\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Inverse Problem\n",
    "\n",
    "Compute $\\pi_{\\Lambda}^u$ and $\\pi_{\\Lambda}^{u,n}$\n",
    "\n",
    "Observed pdf is $\\pi_{\\mathcal{D}} \\sim N(0.3,0.1^2)$\n",
    "\n",
    "Guess is $\\lambda_1\\sim N(0,0.1)$, $\\lambda_2\\sim N(0,0.1)$\n",
    "\n",
    "Verify Result of Theorem 4.2:\n",
    "\n",
    "$Q_n(\\lambda)\\to Q(\\lambda)$ in $L^p(\\Lambda)$, $\\pi_{\\Lambda}^{init}\\in L^p(\\mathcal{D})$. If Assumptions 1, 2 hold, $\\{\\pi_{\\mathcal{D}}^{Q_n}\\}$ are uniformly integrable in $L^p(\\mathcal{D})$, then \n",
    "\\begin{equation}\n",
    "        \\pi_{\\Lambda}^{u,n}(\\lambda) \\to \\pi_{\\Lambda}^{u}(\\lambda) \\text{ in } L^p(\\Lambda) \n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:23:58.022374Z",
     "start_time": "2021-07-01T03:23:58.013431Z"
    }
   },
   "outputs": [],
   "source": [
    "def rejection_sampling(r):\n",
    "    N = r.size # size of proposal sample set\n",
    "    check = np.random.uniform(low=0,high=1,size=N) # create random uniform weights to check r against\n",
    "    M = np.max(r)\n",
    "    new_r = r/M     # normalize weights \n",
    "    idx = np.where(new_r>=check)[0] # rejection criterion\n",
    "    return idx\n",
    "\n",
    "def pdf_obs(x):\n",
    "    return norm.pdf(x, loc=0.1, scale=0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:23:58.597156Z",
     "start_time": "2021-07-01T03:23:58.026745Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#### Use plot to show the difference between the pushforward of the init and the observed #####\n",
    "plt.figure()\n",
    "xx = np.linspace(-1,1,100)\n",
    "plt.plot(xx,pdf_obs(xx),label=\"$\\pi_{\\mathcal{D}}^{obs}$\")\n",
    "plt.plot(xx,pfprior_dens(xx), label=\"$\\pi_{\\mathcal{D}}^{Q(init)}$\")\n",
    "plt.xlabel(\"$\\mathcal{D}$\")\n",
    "plt.legend();\n",
    "# fig.savefig(\"images/2obs_pushforward.png\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Verify Assumption 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:24:12.308419Z",
     "start_time": "2021-07-01T03:23:58.601941Z"
    }
   },
   "outputs": [],
   "source": [
    "def Meanr(n):\n",
    "    if n==0:\n",
    "        pfprior_sample = Qexact(lam1_seed,lam2_seed)\n",
    "        r = pdf_obs(pfprior_sample)/pfprior_dens(pfprior_sample)\n",
    "    else:\n",
    "        pfprior_sample_n = Q(n,lam1_seed,lam2_seed)\n",
    "        r = pdf_obs(pfprior_sample_n)/pfprior_dens_n(n,pfprior_sample_n)\n",
    "    return np.mean(r)\n",
    "\n",
    "Expect_r = np.zeros(6)\n",
    "for i in range(6):\n",
    "    Expect_r[i] = Meanr(i)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:24:12.334804Z",
     "start_time": "2021-07-01T03:24:12.312253Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Expected ratio for verifying Assumption 2\n",
      "[1.00179843 1.00179844 1.00197    1.00197    1.00197088]\n"
     ]
    }
   ],
   "source": [
    "print('Expected ratio for verifying Assumption 2')\n",
    "print(Expect_r[1:])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Verify Theorem 4.2\n",
    "\n",
    "Print out Monte Carlo Approximation of $\\|\\pi_{\\Lambda}^{u,n}(\\lambda)-\\pi_{\\Lambda}^u(\\lambda)\\|_{L^2(\\Lambda)} $ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align*}\n",
    "\\|\\pi_{\\Lambda}^{u,n}(\\lambda)-\\pi_{\\Lambda}^u(\\lambda)\\|^2_{L^2(\\Lambda)} &= \\int (\\pi^i(\\lambda))^2 (r_n(\\lambda) - r(\\lambda))^2\\, d\\mu_{\\Lambda}\\\\\n",
    "&= \\mathbb{E}_i (\\pi^i(\\lambda)(r_n(\\lambda) - r(\\lambda))^2)\\\\\n",
    "&\\approx \\frac{1}{N} \\sum_{j=1}^N \\pi^i(\\lambda^{(j)})(r_n(\\lambda^{(j)}) -r(\\lambda^{(j)}) )^2\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:24:15.947159Z",
     "start_time": "2021-07-01T03:24:12.365803Z"
    }
   },
   "outputs": [],
   "source": [
    "##### Load data for Fig 5 #####\n",
    "# Print out Monte Carlo Approximation of $\\|\\pi_{\\Lambda}^{u,n}(\\lambda)-\\pi_{\\Lambda}^u(\\lambda)\\|_{L^2(\\Lambda)} $\n",
    "init_eval = np.zeros(int(1E4))\n",
    "for i in range(int(1E4)):\n",
    "    init_eval[i] = norm.pdf(lam1_seed[i], loc=0.1, scale=0.1)*norm.pdf(lam2_seed[i], loc=0.1, scale=0.1)\n",
    "\n",
    "r = np.zeros(int(1E4))\n",
    "for i in range(proc_size):\n",
    "    filename = os.path.join(os.getcwd(), \"Data\", \"r_\") + str(i) + \".mat\" \n",
    "    partial_data = sio.loadmat(filename)\n",
    "    r += partial_data['r'].reshape(int(1E4))\n",
    "\n",
    "rn = np.zeros((6,int(1E4)))\n",
    "for i in range(6):\n",
    "    for j in range(proc_size):\n",
    "        filename = os.path.join(os.getcwd(), \"Data\", \"r\")  + str(i+1) + '_' + str(j) + \".mat\" \n",
    "        partial_data = sio.loadmat(filename)\n",
    "        rn[i,:] += partial_data['r'].reshape(int(1E4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:24:16.047676Z",
     "start_time": "2021-07-01T03:24:15.952794Z"
    }
   },
   "outputs": [],
   "source": [
    "error_Update = np.zeros(5)\n",
    "\n",
    "for i in range(5):\n",
    "    error_Update[i] = (np.mean(init_eval*(rn[i,:] - r)**2))**(1/2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:24:16.065372Z",
     "start_time": "2021-07-01T03:24:16.051464Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "L^2 Error for Inverse Problem\n",
      "\n",
      "[4.25879362e-01 4.25878780e-01 5.37649557e-03 5.37799511e-03 3.87794081e-04]\n"
     ]
    }
   ],
   "source": [
    "np.set_printoptions(linewidth=110) \n",
    "print('L^2 Error for Inverse Problem',end='\\n\\n')\n",
    "print(error_Update)   "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Figure 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-07-01T03:24:16.648365Z",
     "start_time": "2021-07-01T03:24:16.070249Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "###########################################\n",
    "################ Figure 5 #################\n",
    "###########################################\n",
    "fig = plt.figure()\n",
    "plt.xlim([0,6])\n",
    "plt.semilogy([1,2,3,4,5], error_Update, '-s')#, label='$L^2(\\Lambda)$ error')\n",
    "plt.xlabel('Order of PCE (n)')\n",
    "plt.ylabel('$L^2$'+' Error in Update')\n",
    "# fig.savefig(\"images/2inverse_error.png\")\n",
    "fig.savefig(\"images/Fig5.png\")"
   ]
  }
 ],
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