FinalProject_2021_2022.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "3fe94688",
   "metadata": {
    "id": "3fe94688"
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.linalg import solve_triangular\n",
    "from scipy.linalg import lu\n",
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8184faab",
   "metadata": {
    "id": "8184faab"
   },
   "source": [
    "Consider the following one-dimensional PDE:\n",
    "$$\n",
    "-u_{xx}(x) = f(x)\\quad\\mathrm{ in }\\ \\Omega = (0, \\pi)\n",
    "$$\n",
    "$$\n",
    "u(x) = 0, \\quad\\mathrm{ on }\\ \\partial\\Omega = \\{0, \\pi\\}\n",
    "$$\n",
    "\n",
    "Given the following $4^{th}$ order finite difference approximation of the second order derivative:\n",
    "\n",
    "$$u_{xx}(x_i) = \\frac{-u_{i-2}+16u_{i-1}-30u_i+16u_{i+1}-u_{i+2}}{12h^2}$$\n",
    "\n",
    "Implement a function that given the domain interval, the forcing function, the number of discretization points, the boundary conditions, returns the matrix $A$ and the the right hand side $b$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "1c562f13",
   "metadata": {
    "id": "1c562f13"
   },
   "outputs": [],
   "source": [
    "def finDif(omega,f,n,bc):\n",
    "    h = (omega[1] - omega[0]) / (n-1)\n",
    "    \n",
    "    # Part for solving the matrix = A\n",
    "    coeff_1 = 1*np.ones((n-2,))\n",
    "    coeff_2 = -16*np.ones((n-1,))\n",
    "    coeff_3 = 30*np.ones((n,))\n",
    "    \n",
    "    #A = np.zeros((n,n))\n",
    "    \n",
    "    A = (np.diag(coeff_1, -2) + np.diag(coeff_2, -1) + np.diag(coeff_3, 0)\n",
    "        + np.diag(coeff_2, +1) + np.diag(coeff_1, +2))\n",
    "    \n",
    "    A = A/(12.*(h**2))\n",
    "    \n",
    "    # Part for solving the RHS = b\n",
    "    x = np.linspace(omega[0], omega[1], n)\n",
    "    b = f(x)\n",
    "    \n",
    "    # Boundary conditions\n",
    "    A[0,:] = 0\n",
    "    A[:,0] = 0\n",
    "    A[-1,:] = 0\n",
    "    A[:,-1] = 0\n",
    "    A[0,0] = A[-1,-1] = 1\n",
    "    b[0] = bc[0]/(h**2)\n",
    "    b[-1] = bc[1]/(h**2)\n",
    "    \n",
    "    return A, b"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c21bb21e",
   "metadata": {
    "id": "c21bb21e"
   },
   "source": [
    "Call the function using:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "23af8831",
   "metadata": {
    "id": "23af8831"
   },
   "outputs": [],
   "source": [
    "omega = [0,np.pi]\n",
    "f = lambda x : np.sin(x)\n",
    "n=100\n",
    "bc = [0,0]\n",
    "A, b = finDif(omega, f, n, bc)\n",
    "\n",
    "#print(A)\n",
    "#print(b)\n",
    "\n",
    "# Solve the linear system using numpy\n",
    "xx = np.linspace(omega[0], omega[1], n)\n",
    "#fx = np.sin(xx)\n",
    "#A1 = A.copy()\n",
    "#u = np.linalg.solve(A1, fx)\n",
    "u_numpy = np.linalg.solve(A, b)\n",
    "u_ex = np.sin(xx)\n",
    "\n",
    "# Plot the FD and exact solution\n",
    "#_ = plt.plot(xx,u,'ro',label='estimated solution')\n",
    "#_ = plt.plot(xx,u_ex,'go',label='exact solution')\n",
    "#plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4f8a7c6f",
   "metadata": {
    "id": "4f8a7c6f"
   },
   "source": [
    "Implement two functions that compute the LU and the Cholesky factorization of the system matrix $A$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "47468a37",
   "metadata": {
    "id": "47468a37"
   },
   "outputs": [],
   "source": [
    "def LU(A):\n",
    "    A = A.copy()\n",
    "    N=len(A)\n",
    "    \n",
    "    for k in range(N-1):\n",
    "        if (abs(A[k,k]) < 1e-15):\n",
    "            raise RuntimeError(\"Null pivot\")\n",
    "            \n",
    "        A[k+1:N,k] /= A[k,k]\n",
    "        \n",
    "        for j in range(k+1,N):\n",
    "            A[k+1:N,j] -= A[k+1:N,k]*A[k,j]\n",
    "    \n",
    "    L=np.tril(A)\n",
    "    \n",
    "    for i in range(N):\n",
    "        L[i,i]=1.0\n",
    "        \n",
    "    U = np.triu(A)\n",
    "    return L, U\n",
    "\n",
    "L, U = LU(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "193ebfed",
   "metadata": {
    "id": "193ebfed"
   },
   "outputs": [],
   "source": [
    "def cholesky(A):\n",
    "    A = A.copy()\n",
    "    N = len(A)\n",
    "    \n",
    "    for k in range(N-1):\n",
    "        A[k,k] = np.sqrt(A[k,k])\n",
    "        A[k+1:N,k] = A[k+1:N,k]/A[k,k]\n",
    "        \n",
    "        for j in range(k+1,N):\n",
    "            A[j:N,j] = A[j:N,j] - A[j:N,k]*A[j,k]\n",
    "        \n",
    "    A[-1,-1] = np.sqrt(A[-1,-1])\n",
    "    L=np.tril(A)\n",
    "    return L, L.transpose()\n",
    "\n",
    "HT, H = cholesky(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "41eb8436",
   "metadata": {
    "id": "41eb8436"
   },
   "source": [
    "Implement forward and backward substitution functions to exploit the developed factorization methods to solve the derived linear system of equations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "019cf0df",
   "metadata": {
    "id": "019cf0df"
   },
   "outputs": [],
   "source": [
    "def L_solve(L,rhs):\n",
    "    x = np.zeros_like(rhs)\n",
    "    N = len(L)\n",
    "        \n",
    "    x[0] = rhs[0]/L[0,0]\n",
    "    \n",
    "    for i in range(1,N):\n",
    "        x[i] = (rhs[i] - np.dot(L[i, 0:i], x[0:i]))/L[i,i]\n",
    "        \n",
    "    return x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "26ce17c4",
   "metadata": {
    "id": "26ce17c4"
   },
   "outputs": [],
   "source": [
    "def U_solve(U,rhs):\n",
    "    x = np.zeros_like(rhs)\n",
    "    N = len(U)\n",
    "    \n",
    "    x[-1] = rhs[-1]/L[-1,-1]\n",
    "    \n",
    "    for i in reversed(range(N-1)):\n",
    "        x[i] = (rhs[i] - np.dot(U[i, i+1:N], x[i+1:N]))/U[i,i]\n",
    "\n",
    "    return x"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "08ac9ba1",
   "metadata": {
    "id": "08ac9ba1"
   },
   "source": [
    "Solve the derived linear system using the implemented functions and plot the computed solution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "a7f45d96-c397-48fc-975a-83e6d615c76c",
   "metadata": {
    "id": "7c1de4f3"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Maximum error (numpy): 6.294964549624638e-14\n",
      "Maximum error (analytical): 0.0024541617562829354\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xx = np.linspace(omega[0], omega[1], n)\n",
    "\n",
    "w = L_solve(L,b)\n",
    "u = U_solve(U,w)\n",
    "\n",
    "plt.title(\"LU Factorization\")\n",
    "_ = plt.plot(xx,u,'yo',label='estimated solution')\n",
    "_ = plt.plot(xx,u_ex, 'r.',label='exact solution')\n",
    "_ = plt.plot(xx,u_numpy, 'b',label='numpy solution')\n",
    "plt.legend()\n",
    "\n",
    "print(\"Maximum error (numpy):\", max(abs(u-u_numpy)))\n",
    "print(\"Maximum error (analytical):\", max(abs(u-u_ex)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "c9006941-fdbd-434d-a717-c6d88495e8a2",
   "metadata": {
    "id": "7c1de4f3"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Maximum error (numpy): 2.5646151868841116e-14\n",
      "Maximum error (analytical): 0.0024541617562779117\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xx = np.linspace(omega[0], omega[1], n)\n",
    "\n",
    "w = L_solve(HT,b)\n",
    "u = U_solve(H,w)\n",
    "\n",
    "plt.title(\"Cholesky Factorization\")\n",
    "_ = plt.plot(xx,u,'yo',label='estimated solution')\n",
    "_ = plt.plot(xx,u_ex, 'r.',label='exact solution')\n",
    "_ = plt.plot(xx,u_numpy, 'b',label='numpy solution')\n",
    "plt.legend()\n",
    "\n",
    "print(\"Maximum error (numpy):\", max(abs(u-u_numpy)))\n",
    "print(\"Maximum error (analytical):\", max(abs(u-u_ex)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "62cdfe65",
   "metadata": {
    "id": "62cdfe65"
   },
   "source": [
    "Considering the new domain $\\Omega = (0,1)$ and the forcing term $f(x) = x(1-x)$ with B.C. $u(x) = 0$, on $\\partial \\Omega = {0,1}$ produce a plot and a table where you show the decay of the error w.r.t. the number of grid points.\n",
    "(The analytical solution for the above problems is $u_{an} = \\frac{x^4}{12} - \\frac{x^3}{6} + \\frac{x}{12}$)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "35a88c3b-4bcd-4c20-8e26-0655836cde60",
   "metadata": {
    "id": "91212afb"
   },
   "outputs": [],
   "source": [
    "def error_analysis(f,bc,omega,n_range,f_exact):\n",
    "    error=[]\n",
    "    for i in n_range:\n",
    "        x_vals = np.linspace(omega[0], omega[1], i)\n",
    "        #u_ex = x_vals**4/12. - x_vals**3/6. + x_vals/12.\n",
    "        u_ex = f_exact(x_vals)\n",
    "        A_i, b_i = finDif(omega, f, i, bc)\n",
    "        \n",
    "        L_i, U_i = LU(A_i)\n",
    "        w_i = L_solve(L_i, b_i)\n",
    "        u_i = U_solve(U_i, w_i)\n",
    "        \n",
    "        error.append(np.linalg.norm(np.abs(u_ex - u_i), 2))\n",
    "        \n",
    "    return error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "432df4fc-dd3f-4164-af80-0837c4111ca2",
   "metadata": {
    "id": "91212afb"
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>n</th>\n",
       "      <th>error</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>10</td>\n",
       "      <td>0.001936</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>20</td>\n",
       "      <td>0.001419</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>30</td>\n",
       "      <td>0.001167</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>40</td>\n",
       "      <td>0.001013</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>50</td>\n",
       "      <td>0.000908</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>60</td>\n",
       "      <td>0.000829</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>70</td>\n",
       "      <td>0.000768</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>80</td>\n",
       "      <td>0.000719</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>90</td>\n",
       "      <td>0.000678</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>100</td>\n",
       "      <td>0.000643</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>110</td>\n",
       "      <td>0.000614</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>120</td>\n",
       "      <td>0.000588</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>130</td>\n",
       "      <td>0.000565</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>140</td>\n",
       "      <td>0.000544</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>150</td>\n",
       "      <td>0.000526</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>160</td>\n",
       "      <td>0.000509</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>170</td>\n",
       "      <td>0.000494</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>180</td>\n",
       "      <td>0.000480</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>190</td>\n",
       "      <td>0.000467</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>200</td>\n",
       "      <td>0.000455</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>210</td>\n",
       "      <td>0.000444</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>220</td>\n",
       "      <td>0.000434</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>230</td>\n",
       "      <td>0.000425</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>240</td>\n",
       "      <td>0.000416</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>250</td>\n",
       "      <td>0.000407</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>260</td>\n",
       "      <td>0.000400</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>270</td>\n",
       "      <td>0.000392</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>280</td>\n",
       "      <td>0.000385</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>290</td>\n",
       "      <td>0.000378</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>300</td>\n",
       "      <td>0.000372</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>30</th>\n",
       "      <td>310</td>\n",
       "      <td>0.000366</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>31</th>\n",
       "      <td>320</td>\n",
       "      <td>0.000360</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>32</th>\n",
       "      <td>330</td>\n",
       "      <td>0.000355</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>33</th>\n",
       "      <td>340</td>\n",
       "      <td>0.000349</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>34</th>\n",
       "      <td>350</td>\n",
       "      <td>0.000344</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>35</th>\n",
       "      <td>360</td>\n",
       "      <td>0.000340</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>36</th>\n",
       "      <td>370</td>\n",
       "      <td>0.000335</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>37</th>\n",
       "      <td>380</td>\n",
       "      <td>0.000331</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>38</th>\n",
       "      <td>390</td>\n",
       "      <td>0.000326</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>39</th>\n",
       "      <td>400</td>\n",
       "      <td>0.000322</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>40</th>\n",
       "      <td>410</td>\n",
       "      <td>0.000318</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>41</th>\n",
       "      <td>420</td>\n",
       "      <td>0.000314</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>42</th>\n",
       "      <td>430</td>\n",
       "      <td>0.000311</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>43</th>\n",
       "      <td>440</td>\n",
       "      <td>0.000307</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>44</th>\n",
       "      <td>450</td>\n",
       "      <td>0.000304</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>45</th>\n",
       "      <td>460</td>\n",
       "      <td>0.000300</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>46</th>\n",
       "      <td>470</td>\n",
       "      <td>0.000297</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>47</th>\n",
       "      <td>480</td>\n",
       "      <td>0.000294</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>48</th>\n",
       "      <td>490</td>\n",
       "      <td>0.000291</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49</th>\n",
       "      <td>500</td>\n",
       "      <td>0.000288</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      n     error\n",
       "0    10  0.001936\n",
       "1    20  0.001419\n",
       "2    30  0.001167\n",
       "3    40  0.001013\n",
       "4    50  0.000908\n",
       "5    60  0.000829\n",
       "6    70  0.000768\n",
       "7    80  0.000719\n",
       "8    90  0.000678\n",
       "9   100  0.000643\n",
       "10  110  0.000614\n",
       "11  120  0.000588\n",
       "12  130  0.000565\n",
       "13  140  0.000544\n",
       "14  150  0.000526\n",
       "15  160  0.000509\n",
       "16  170  0.000494\n",
       "17  180  0.000480\n",
       "18  190  0.000467\n",
       "19  200  0.000455\n",
       "20  210  0.000444\n",
       "21  220  0.000434\n",
       "22  230  0.000425\n",
       "23  240  0.000416\n",
       "24  250  0.000407\n",
       "25  260  0.000400\n",
       "26  270  0.000392\n",
       "27  280  0.000385\n",
       "28  290  0.000378\n",
       "29  300  0.000372\n",
       "30  310  0.000366\n",
       "31  320  0.000360\n",
       "32  330  0.000355\n",
       "33  340  0.000349\n",
       "34  350  0.000344\n",
       "35  360  0.000340\n",
       "36  370  0.000335\n",
       "37  380  0.000331\n",
       "38  390  0.000326\n",
       "39  400  0.000322\n",
       "40  410  0.000318\n",
       "41  420  0.000314\n",
       "42  430  0.000311\n",
       "43  440  0.000307\n",
       "44  450  0.000304\n",
       "45  460  0.000300\n",
       "46  470  0.000297\n",
       "47  480  0.000294\n",
       "48  490  0.000291\n",
       "49  500  0.000288"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "omega_new = [0,1]\n",
    "f_new = lambda x : x*(1-x)\n",
    "bc_new = [0,0]\n",
    "\n",
    "n_range = np.arange(10,501,10)\n",
    "f_exact = lambda x : (pow(x,4)/12 - pow(x,3)/6 + x/12)\n",
    "\n",
    "errors = error_analysis(f_new, bc_new, omega_new, n_range,f_exact)\n",
    "\n",
    "df = pd.DataFrame({'n' : n_range, 'error' : errors})  \n",
    "display(df)\n",
    "\n",
    "plt.title(\"Error decay - Absolute Error vs 'n'\")\n",
    "_ = plt.semilogy(n_range, errors, 'ro-')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c8d5002f",
   "metadata": {
    "id": "c8d5002f"
   },
   "source": [
    "Exploit the derived LU factorizations to compute the condition number of the system's matrix $A$ using the original problem formulation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "47754c4c-5421-4cf2-ba8f-4a44a05696b4",
   "metadata": {
    "id": "c25fc1fe"
   },
   "outputs": [],
   "source": [
    "# Power Method Function - needed for Condition Number Solution\n",
    "def PM(A,z0,tol,nmax):\n",
    "    q = z0/np.linalg.norm(z0,2)\n",
    "    it = 0\n",
    "    err = tol + 1.\n",
    "    \n",
    "    while it < nmax and err > tol:\n",
    "        z = np.dot(A,q)\n",
    "        l = np.dot(q.T,z)\n",
    "        err = np.linalg.norm(z-l*q,2)\n",
    "        q = z/np.linalg.norm(z,2)\n",
    "        it += 1\n",
    "    \n",
    "    return l,q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "93612f31-8617-4dae-891b-4fa4cce220a4",
   "metadata": {
    "id": "c25fc1fe"
   },
   "outputs": [],
   "source": [
    "# Inverse Power Method Function - needed for Condition Number Solution\n",
    "def IPM(A,x0,mu,tol,nmax):\n",
    "    I = np.eye(len(A))\n",
    "    M = A - mu*I\n",
    "    L,U = LU(M)\n",
    "    q = x0/np.linalg.norm(x0,2)\n",
    "    err = tol + 1.0\n",
    "    it = 0\n",
    "    \n",
    "    while it < nmax and err > tol:\n",
    "        y = L_solve(L,q)\n",
    "        x = U_solve(U,y)\n",
    "        q = x/np.linalg.norm(x,2)\n",
    "        z = np.dot(A,q)\n",
    "        l = np.dot(q.T,z)\n",
    "        err = np.linalg.norm(z-l*q,2)\n",
    "        it += 1\n",
    "        \n",
    "    #I = np.eye(len(A))\n",
    "    #M = A - mu*I\n",
    "    #P,L,U = lu(M) #scipy's LU factorization\n",
    "    #err = tol + 1.\n",
    "    #it = 0\n",
    "    #q = x0/np.linalg.norm(x0,2)\n",
    "    \n",
    "    #while it < nmax and err > tol:\n",
    "    #    y = solve_triangular(L, np.dot(P.T,q), lower=True)\n",
    "    #    x = solve_triangular(U, y)\n",
    "    #    q = x/np.linalg.norm(x,2)\n",
    "    #    z = np.dot(A,q)\n",
    "    #    l = np.dot(q.T,z)\n",
    "    #    err = np.linalg.norm(z-l*q,2)\n",
    "    #    it += 1\n",
    "    \n",
    "    return l,q   "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "8b222a32-509e-4082-836c-4e7bfaf5f018",
   "metadata": {
    "id": "c25fc1fe"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Condition Number we calculate: 5294.598016123485\n",
      "Condition Number from Numpy: 5294.598016140671\n"
     ]
    }
   ],
   "source": [
    "def condNumb(A):\n",
    "    z0 = np.random.rand(A.shape[0])\n",
    "    mu = 0.0\n",
    "    tol = 1e-7\n",
    "    nmax = 13000\n",
    "    lmax = PM(A,z0,tol,nmax)[0]\n",
    "    lmin = IPM(A,z0,mu,tol,nmax)[0]\n",
    "    condNu = lmax/lmin\n",
    "    \n",
    "    return condNu\n",
    "\n",
    "print(\"Condition Number we calculate:\", condNumb(A))\n",
    "print(\"Condition Number from Numpy:\", np.linalg.cond(A))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2728b49a",
   "metadata": {
    "id": "2728b49a"
   },
   "source": [
    "Implement a preconditioned Conjugant Gradient method to solve the original linear system of equations using an iterative method:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "60dfc640-2639-4e11-9087-50591cef7b60",
   "metadata": {
    "id": "62b83aee"
   },
   "outputs": [],
   "source": [
    "def conjugate_gradient(A, b, P, nmax=len(A), eps=1e-11):\n",
    "    N = len(A)\n",
    "    x = np.zeros_like(b)\n",
    "    r = b - np.dot(A,x)\n",
    "    rho_old = 1.\n",
    "    p_old = np.zeros_like(b)\n",
    "    tol = eps + 1.0\n",
    "    it = 0\n",
    "    \n",
    "    while (it < nmax and tol > eps):\n",
    "        it += 1\n",
    "        z = np.linalg.solve(P,r)\n",
    "        rho = np.dot(r,z)\n",
    "        if (it > 1):\n",
    "            beta = rho/rho_old\n",
    "            p = z + beta*p_old\n",
    "        else:\n",
    "            p = z\n",
    "        \n",
    "        q = np.dot(A,p)\n",
    "        alpha = rho/np.dot(p,q)\n",
    "        \n",
    "        x += p*alpha\n",
    "        r -= q*alpha\n",
    "        p_old = p\n",
    "        rho_old = rho\n",
    "        tol = np.linalg.norm(r,2)\n",
    "\n",
    "    #print(it,tol)\n",
    "\n",
    "    return x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "c6ef1ff6-ae0d-4105-a30f-7e46ffc42674",
   "metadata": {
    "id": "62b83aee"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Maximum error (numpy): 2.5646151868841116e-14\n",
      "Maximum error (analytical): 0.0024541617562779117\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Verifying Conjugate Gradient solution\n",
    "xx = np.linspace(omega[0], omega[1], n)\n",
    "u_ex = np.sin(xx)\n",
    "u_cg = conjugate_gradient(A, b, np.diag(np.diag(A)))\n",
    "\n",
    "plt.title(\"Conjugate Gradient Solution\")\n",
    "_ = plt.plot(xx,u_cg,'yo',label='estimated solution')\n",
    "_ = plt.plot(xx,u_ex, 'r.',label='exact solution')\n",
    "_ = plt.plot(xx,u_numpy, 'b',label='numpy solution')\n",
    "plt.legend()\n",
    "\n",
    "print(\"Maximum error (numpy):\", max(abs(u-u_numpy)))\n",
    "print(\"Maximum error (analytical):\", max(abs(u-u_ex)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8a4cfc02",
   "metadata": {
    "id": "8a4cfc02"
   },
   "source": [
    "Consider the following time dependent variation of the PDE starting from the orginal problem formulation:\n",
    "$$\n",
    "u'(t)-u_{xx} = \\alpha(t)f(x)\n",
    "$$\n",
    "\n",
    "for $t\\in [0,T]$, with $\\alpha(t) = \\cos(t)$ and $T = 6\\pi$\n",
    "\n",
    "Use the same finite difference scheme to derive the semi-discrete formulation and solve it using a forward Euler's method.\n",
    "\n",
    "Plot the time dependent solution solution at $x = \\pi/2$, $x=1$, \n",
    "$x=\\pi$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "6ec202e9-0836-419c-8e8b-167163c19d00",
   "metadata": {
    "id": "3ffe0689"
   },
   "outputs": [],
   "source": [
    "# Forward Euler Method Function\n",
    "def fe(A,b,l,y0,t0,tf,h):\n",
    "    timesteps = np.arange(t0,tf+1e-10,h)\n",
    "    t = timesteps\n",
    "    sol = np.zeros((len(t), len(y0)))\n",
    "    sol[0,:] = y0\n",
    "    \n",
    "    for i in range(1,len(t)):\n",
    "        u_xx = -np.dot(A,sol[i-1])     \n",
    "        l_b = np.dot(l(t[i-1]),b)       \n",
    "        sol[i] = sol[i-1] + h*u_xx + h*l_b\n",
    "    \n",
    "    return sol, timesteps"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "f16e7245-04cf-4744-8156-97e9ea6aa765",
   "metadata": {
    "id": "3ffe0689"
   },
   "outputs": [],
   "source": [
    "t0 = 0\n",
    "tf = 6*np.pi\n",
    "alpha = lambda x: np.cos(x)\n",
    "\n",
    "omega = [0,np.pi] # reusing original omega\n",
    "n=100 # reusing original n\n",
    "\n",
    "# inv of max eigenvalue is taken as a step\n",
    "tol = 1e-7\n",
    "nmax = 13000\n",
    "xx = np.linspace(omega[0], omega[1], n)\n",
    "eigenvalue_max,_ = PM(A,np.ones_like(xx),tol,nmax) \n",
    "h = 1/(eigenvalue_max)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "cd10c7b5-db84-4334-8156-f6919d6ec6e5",
   "metadata": {
    "id": "3ffe0689"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Initial condition: y0 = sin(x)\n",
    "y0 = np.sin(xx)\n",
    "sol,t = fe(A,b,alpha,y0,t0,tf,h)\n",
    "\n",
    "width = omega[1] - omega[0]\n",
    "x_pi_2 = int(np.floor(n/width * np.pi/2.) - 1)\n",
    "x_one = int(np.floor((n/width) - 1))\n",
    "x_pi = int(np.floor((n/width *np.pi)) - 1)\n",
    "\n",
    "plt.title(\"Initial condition: y0 = sin(x)\")\n",
    "plt.plot(t, sol[:,x_one], label='x=1')\n",
    "plt.plot(t, sol[:,x_pi_2], label='x=π/2')\n",
    "plt.plot(t, sol[:,x_pi], label='x=π')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "8c62e1cd-3758-4246-b89b-3655a0cbfcd5",
   "metadata": {
    "id": "3ffe0689"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Initial condition: y0 = 0\n",
    "y0 = np.zeros((n,))\n",
    "sol, t = fe(A,b,alpha,y0,t0,tf,h)\n",
    "\n",
    "width = omega[1] - omega[0]\n",
    "x_pi_2 = int(np.floor(n/width * np.pi/2.) - 1)\n",
    "x_one = int(np.floor((n/width) - 1))\n",
    "x_pi = int(np.floor((n/width *np.pi)) - 1)\n",
    "\n",
    "plt.title(\"Initial condition: y0 = 0\")\n",
    "plt.plot(t, sol[:,x_one], label='x=1')\n",
    "plt.plot(t, sol[:,x_pi_2], label='x=π/2')\n",
    "plt.plot(t, sol[:,x_pi], label='x=π')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "36936121",
   "metadata": {
    "id": "36936121"
   },
   "source": [
    "Given the original $Au = b$ system, implement an algorithm to compute the eigenvalues and eigenvectors of the matrix $A$. Exploit the computed LU factorization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "195097b1-e71b-448b-90a4-de986cb6a394",
   "metadata": {
    "id": "622aadf4"
   },
   "outputs": [],
   "source": [
    "def eigenvalue_LU(A,tol,nmax):\n",
    "    B = A.copy()\n",
    "    eigval_old = np.diag(B)\n",
    "    err = tol+1.0\n",
    "    it = 0\n",
    "    \n",
    "    while it < nmax and err > tol:\n",
    "        L,U = LU(B)\n",
    "        B = U@L\n",
    "        eigval_new = np.diag(B)  \n",
    "        err = np.linalg.norm(eigval_new - eigval_old,2)\n",
    "        it += 1\n",
    "        eigval_old = eigval_new\n",
    "        \n",
    "    return eigval_new\n",
    "\n",
    "def eigenvector_LU(A,eigval,tol,nmax,nmax_ipm):\n",
    "    eigvec = np.zeros_like(A)\n",
    "    eigval_new = np.zeros_like(eigval)\n",
    "    q = np.random.rand(A.shape[0])\n",
    "\n",
    "    for i in range(len(eigval)):\n",
    "        eigval_val, eigvec_val = IPM(A,q,eigval[i]+tol,tol,nmax_ipm)\n",
    "        eigval_new[i] = eigval_val\n",
    "        eigvec[:,i] = eigvec_val\n",
    "         \n",
    "    return eigval_new, eigvec\n",
    "\n",
    "nmax = 2300\n",
    "nmax_ipm = 13000\n",
    "tol = 1e-7\n",
    "eigenvalues = eigenvalue_LU(A,tol,nmax)\n",
    "eigvals_new, eigenvectors = eigenvector_LU(A,eigenvalues,tol,nmax,nmax_ipm)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "f84ff937-9054-44c7-9cdc-d264ea3ce33d",
   "metadata": {
    "id": "622aadf4"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean Eigenvalue Error = 1.5101853101384678e-12\n",
      "Mean Eigenvector Norm Error = 1.414213597640233\n"
     ]
    }
   ],
   "source": [
    "# Compare with Numpy\n",
    "eval_np, evec_np = np.linalg.eig(A)\n",
    "print(\"Mean Eigenvalue Error =\", abs(np.mean(eval_np-eigenvalues)))\n",
    "\n",
    "evec_err = np.zeros_like(eval_np)\n",
    "for i in range(len(eval_np)):\n",
    "    evec_err[i] = np.linalg.norm((eigenvectors[:,i] - evec_np[:,i]), 2)\n",
    "\n",
    "print(\"Mean Eigenvector Norm Error =\", abs(np.mean(evec_err)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "85d5f64e",
   "metadata": {
    "id": "85d5f64e"
   },
   "source": [
    "Compute the inverse of the matrix A exploiting the derived LU factorization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "6ad7199f",
   "metadata": {
    "id": "6ad7199f"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Maximum error in comparison to numpy: 1.2836953722228372e-15\n"
     ]
    }
   ],
   "source": [
    "def inverse_matrix_LU(A):\n",
    "    L,U = LU(A)\n",
    "    N = len(A)\n",
    "    A_inv = np.zeros_like(A)\n",
    "    I = np.identity(A.shape[0])\n",
    "    \n",
    "    for i in range(N):\n",
    "        q = L_solve(L,I[:,i])\n",
    "        A_inv_i = U_solve(U,q)\n",
    "        A_inv[:,i] = A_inv_i\n",
    "    \n",
    "    return A_inv\n",
    "\n",
    "# Compare inverse to that computed by Numpy\n",
    "A_inv_LU = inverse_matrix_LU(A)\n",
    "A_inv_numpy = np.linalg.inv(A)\n",
    "print(\"Maximum error in comparison to numpy:\", max(np.abs(A_inv_LU - A_inv_numpy).max(axis=0)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cb22566e",
   "metadata": {
    "id": "cb22566e"
   },
   "source": [
    "Consider the following Cauchy problem\n",
    "$$\n",
    "\\begin{cases}\n",
    "y'= -ty^2 \\quad 0\\le t \\le 2\\\\\n",
    "y(0) = 1\n",
    "\\end{cases}\n",
    "$$\n",
    "Implement a Backward Euler's method in a suitable function and solve the resulting non-linear equation using a Newton's method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "211a5127-bf80-45b1-ba7f-429561373e1b",
   "metadata": {
    "id": "3184e358"
   },
   "outputs": [],
   "source": [
    "# Newton's Method function\n",
    "def newton(f,f_prime,x0,eps,n_max):\n",
    "    assert np.abs(f_prime(x0)) > 1e-16\n",
    "    err = np.abs(f(x0))\n",
    "    errors = [err]\n",
    "    it = 0\n",
    "    x = x0\n",
    "    \n",
    "    while (err > eps and it < n_max):\n",
    "        qk = f_prime(x)\n",
    "        if np.abs(qk) < 1e-12:\n",
    "            raise RuntimeError(\"f_prime(x) is close to zero\")\n",
    "        \n",
    "        x_new = x - f(x)/qk \n",
    "        err = np.abs(f(x_new))\n",
    "        \n",
    "        x = x_new\n",
    "        errors.append(err)\n",
    "        it += 1\n",
    "            \n",
    "        #plt.semilogy(errors)\n",
    "\n",
    "    return x, errors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "2209bac8-63eb-4078-bd29-c41790b272dc",
   "metadata": {
    "id": "3184e358"
   },
   "outputs": [],
   "source": [
    "# Backward Euler Method Function\n",
    "def be(y0,t0,tf,h):\n",
    "    timesteps = np.arange(t0, tf+1e-10, h)\n",
    "    t = timesteps\n",
    "    sol = np.zeros_like(timesteps)\n",
    "    sol[0] = y0\n",
    "    eps = 1e-5\n",
    "    n_max = 1000\n",
    "    \n",
    "    for i in range(1, len(sol)):\n",
    "        f = lambda y: t[i]*h*y**2 + y - sol[i-1]\n",
    "        df = lambda y: 2*t[i]*h*y + 1\n",
    "        sol[i],_ = newton(f, df, sol[i-1], eps, n_max)\n",
    "\n",
    "    return sol, timesteps"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "7ca33ca9-46a6-43b6-8392-115e144b7b75",
   "metadata": {
    "id": "3184e358"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f6e42a554f0>"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot solution of Cauchy Problem\n",
    "\n",
    "t0 = 0.0\n",
    "tf = 2.0\n",
    "y0 = 1.0\n",
    "\n",
    "y,t = be(y0, t0, tf, 0.05)\n",
    "y_ex = 2.0/(t*t + 2.0)\n",
    "\n",
    "plt.title(\"Solution using Newton's Method and Backward Euler\")\n",
    "_ = plt.plot(t, y, 'g', label='estimated solution')\n",
    "_ = plt.plot(t, y_ex, 'r*', label='exact solution')\n",
    "plt.legend()"
   ]
  }
 ],
 "metadata": {
  "colab": {
   "name": "FinalProject-2021-2022.ipynb",
   "provenance": []
  },
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
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