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68eae05c · Benjamin Berkels · 78221d61 · 68eae05c
Commit 68eae05c authored 5 years ago by Benjamin Berkels
--- a/exp-355.ipynb
+++ b/exp-355.ipynb
+{
+  "cells": [
+    {
+      "cell_type": "markdown",
+      "metadata": {},
+      "source": [
+        "### IPython notebook for Example 3.5.5 from the lecture"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [
+        "import numpy as np\n",
+        "import matplotlib.pyplot as plt\n",
+        "from IPython.display import set_matplotlib_formats, display, Math\n",
+        "\n",
+        "\n",
+        "def k(s, t, d):\n",
+        "    return d / np.power(d**2 + (s-t)**2, 3/2)\n",
+        "\n",
+        "\n",
+        "d = 0.1\n",
+        "n = 80\n",
+        "h = 1/n\n",
+        "\n",
+        "display(Math(r'\\text{Create }A\\text{ from Example 1.7.1, }\\xi_j=\\begin{cases}2&1\\leq j\\leq40\\\\1&41\\leq j\\leq80\\end{cases},\\ y=A\\xi'))\n",
+        "A = np.zeros((n, n))\n",
+        "for i in range(n):\n",
+        "    for j in range(n):\n",
+        "        A[i, j] = h * k((i+0.5)*h, (j+0.5)*h, d)\n",
+        "\n",
+        "xi = np.ones(n)\n",
+        "xi[:n//2] = 2\n",
+        "y = np.matmul(A, xi)"
+      ],
+      "outputs": [],
+      "execution_count": null
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [
+        "display(Math(r'\\tilde y= y+\\delta y'))\n",
+        "np.random.seed(0)\n",
+        "y_tilde = y + 0.02*(np.random.rand(n)-0.5)\n",
+        "epsilon_a = np.linalg.norm(y_tilde-y)\n",
+        "print('epsilon_a=||y_tilde-y|| = {:.3f}'.format(epsilon_a))"
+      ],
+      "outputs": [],
+      "execution_count": null
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [
+        "discrepancy = np.zeros(100)\n",
+        "alphas = np.zeros(100)\n",
+        "xi_alphas = np.zeros((100, n))\n",
+        "for i in range(1, 101):\n",
+        "    alpha = 0.005*i\n",
+        "    A_Tikh = np.concatenate((A, np.diag(alpha*np.ones(n))), axis=0)\n",
+        "    y_tilde_Tikh = np.concatenate((y_tilde, np.zeros(n)), axis=0)\n",
+        "    xi_alpha = np.linalg.lstsq(A_Tikh, y_tilde_Tikh, rcond=0)[0]\n",
+        "    xi_alphas[i-1, :] = xi_alpha\n",
+        "    alphas[i-1] = alpha\n",
+        "    discrepancy[i-1] = np.linalg.norm(np.matmul(A, xi_alpha)-y_tilde)\n",
+        "\n",
+        "set_matplotlib_formats('svg')\n",
+        "plt.title(\"Tikhonov, discrepancy\")\n",
+        "plt.xlabel(r'$\\alpha$')\n",
+        "plt.ylabel(r'$||A\\tilde \\xi_{\\alpha}- \\tilde y||_2$')\n",
+        "plt.plot(alphas, discrepancy, 'k+')\n",
+        "plt.hlines(epsilon_a, xmin=alphas[0], xmax=alphas[-1])\n",
+        "plt.show()\n",
+        "plt.close()"
+      ],
+      "outputs": [],
+      "execution_count": null
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [
+        "alpha_index = np.argmin(np.abs(discrepancy-epsilon_a))\n",
+        "print('alpha_d = {}'.format(alphas[alpha_index]))\n",
+        "\n",
+        "j = np.arange(1, n+1)\n",
+        "plt.title(\"Tikhonov, alpha_d = {}\".format(alphas[alpha_index]))\n",
+        "plt.plot(j, xi_alphas[alpha_index, :], 'kx')\n",
+        "plt.show()\n",
+        "plt.close()"
+      ],
+      "outputs": [],
+      "execution_count": null
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [
+        "# CGNE\n",
+        "xi_tilde = np.zeros(n)\n",
+        "xi_ks = np.zeros((101, n))\n",
+        "# xi_tilde is zero, so the initial residuum is the input data.\n",
+        "r = y_tilde.copy()\n",
+        "d = np.matmul(A.T, r)\n",
+        "p = d.copy()\n",
+        "p_norm = np.linalg.norm(p)\n",
+        "discrepancy = np.zeros(101)\n",
+        "discrepancy[0] = np.linalg.norm(np.matmul(A, xi_tilde)-y_tilde)\n",
+        "for k in range(1, 101):\n",
+        "    q = np.matmul(A, d)\n",
+        "    beta = (np.linalg.norm(p)/np.linalg.norm(q))**2\n",
+        "    xi_tilde += beta * d\n",
+        "    r += -beta*q\n",
+        "    p = np.matmul(A.T, r)\n",
+        "    p_norm_new = np.linalg.norm(p)\n",
+        "    gamma = (p_norm_new/p_norm)**2\n",
+        "    p_norm = p_norm_new\n",
+        "    d = p + gamma*d\n",
+        "    discrepancy[k] = np.linalg.norm(np.matmul(A, xi_tilde)-y_tilde)\n",
+        "    xi_ks[k, :] = xi_tilde\n",
+        "\n",
+        "plt.title(\"CGNE, discrepancy\")\n",
+        "plt.xlabel(r'$k$')\n",
+        "plt.ylabel(r'$||A\\tilde \\xi_k- \\xi||_2$')\n",
+        "plt.yscale('log')\n",
+        "plt.plot(discrepancy, 'k+')\n",
+        "plt.hlines(epsilon_a, xmin=0, xmax=discrepancy.shape[0])\n",
+        "plt.show()\n",
+        "plt.close()"
+      ],
+      "outputs": [],
+      "execution_count": null
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [
+        "k_opt = np.argmin(np.abs(discrepancy-epsilon_a))\n",
+        "plt.title(\"CGNE, k_opt = {}\".format(k_opt))\n",
+        "plt.plot(j, xi_ks[k_opt, :], 'kx')\n",
+        "plt.show()\n",
+        "plt.close()\n"
+      ],
+      "outputs": [],
+      "execution_count": null
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [],
+      "outputs": [],
+      "execution_count": null
+    }
+  ],
+  "metadata": {
+    "anaconda-cloud": {},
+    "kernelspec": {
+      "display_name": "Python 3",
+      "language": "python",
+      "name": "python3"
+    },
+    "language_info": {
+      "codemirror_mode": {
+        "name": "ipython",
+        "version": 3
+      },
+      "file_extension": ".py",
+      "mimetype": "text/x-python",
+      "name": "python",
+      "nbconvert_exporter": "python",
+      "pygments_lexer": "ipython3",
+      "version": "3.6.1"
+    }
+  },
+  "nbformat": 4,
+  "nbformat_minor": 1
+}
\ No newline at end of file
+%% Cell type:markdown id: tags:
+### IPython notebook for Example 3.5.5 from the lecture
+%% Cell type:code id: tags:
+``` python
+import numpy as np
+import matplotlib.pyplot as plt
+from IPython.display import set_matplotlib_formats, display, Math
+def k(s, t, d):
+    return d / np.power(d**2 + (s-t)**2, 3/2)
+d = 0.1
+n = 80
+h = 1/n
+display(Math(r'\text{Create }A\text{ from Example 1.7.1, }\xi_j=\begin{cases}2&1\leq j\leq40\\1&41\leq j\leq80\end{cases},\ y=A\xi'))
+A = np.zeros((n, n))
+for i in range(n):
+    for j in range(n):
+        A[i, j] = h * k((i+0.5)*h, (j+0.5)*h, d)
+xi = np.ones(n)
+xi[:n//2] = 2
+y = np.matmul(A, xi)
+```
+%% Cell type:code id: tags:
+``` python
+display(Math(r'\tilde y= y+\delta y'))
+np.random.seed(0)
+y_tilde = y + 0.02*(np.random.rand(n)-0.5)
+epsilon_a = np.linalg.norm(y_tilde-y)
+print('epsilon_a=||y_tilde-y|| = {:.3f}'.format(epsilon_a))
+```
+%% Cell type:code id: tags:
+``` python
+discrepancy = np.zeros(100)
+alphas = np.zeros(100)
+xi_alphas = np.zeros((100, n))
+for i in range(1, 101):
+    alpha = 0.005*i
+    A_Tikh = np.concatenate((A, np.diag(alpha*np.ones(n))), axis=0)
+    y_tilde_Tikh = np.concatenate((y_tilde, np.zeros(n)), axis=0)
+    xi_alpha = np.linalg.lstsq(A_Tikh, y_tilde_Tikh, rcond=0)[0]
+    xi_alphas[i-1, :] = xi_alpha
+    alphas[i-1] = alpha
+    discrepancy[i-1] = np.linalg.norm(np.matmul(A, xi_alpha)-y_tilde)
+set_matplotlib_formats('svg')
+plt.title("Tikhonov, discrepancy")
+plt.xlabel(r'$\alpha$')
+plt.ylabel(r'$||A\tilde \xi_{\alpha}- \tilde y||_2$')
+plt.plot(alphas, discrepancy, 'k+')
+plt.hlines(epsilon_a, xmin=alphas[0], xmax=alphas[-1])
+plt.show()
+plt.close()
+```
+%% Cell type:code id: tags:
+``` python
+alpha_index = np.argmin(np.abs(discrepancy-epsilon_a))
+print('alpha_d = {}'.format(alphas[alpha_index]))
+j = np.arange(1, n+1)
+plt.title("Tikhonov, alpha_d = {}".format(alphas[alpha_index]))
+plt.plot(j, xi_alphas[alpha_index, :], 'kx')
+plt.show()
+plt.close()
+```
+%% Cell type:code id: tags:
+``` python
+# CGNE
+xi_tilde = np.zeros(n)
+xi_ks = np.zeros((101, n))
+# xi_tilde is zero, so the initial residuum is the input data.
+r = y_tilde.copy()
+d = np.matmul(A.T, r)
+p = d.copy()
+p_norm = np.linalg.norm(p)
+discrepancy = np.zeros(101)
+discrepancy[0] = np.linalg.norm(np.matmul(A, xi_tilde)-y_tilde)
+for k in range(1, 101):
+    q = np.matmul(A, d)
+    beta = (np.linalg.norm(p)/np.linalg.norm(q))**2
+    xi_tilde += beta * d
+    r += -beta*q
+    p = np.matmul(A.T, r)
+    p_norm_new = np.linalg.norm(p)
+    gamma = (p_norm_new/p_norm)**2
+    p_norm = p_norm_new
+    d = p + gamma*d
+    discrepancy[k] = np.linalg.norm(np.matmul(A, xi_tilde)-y_tilde)
+    xi_ks[k, :] = xi_tilde
+plt.title("CGNE, discrepancy")
+plt.xlabel(r'$k$')
+plt.ylabel(r'$||A\tilde \xi_k- \xi||_2$')
+plt.yscale('log')
+plt.plot(discrepancy, 'k+')
+plt.hlines(epsilon_a, xmin=0, xmax=discrepancy.shape[0])
+plt.show()
+plt.close()
+```
+%% Cell type:code id: tags:
+``` python
+k_opt = np.argmin(np.abs(discrepancy-epsilon_a))
+plt.title("CGNE, k_opt = {}".format(k_opt))
+plt.plot(j, xi_ks[k_opt, :], 'kx')
+plt.show()
+plt.close()
+```
+%% Cell type:code id: tags:
+``` python
+```