seminars/seminar12/seminar12.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Итерационные методы для частичной задачи на  собственные значения"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## PINVIT\n",
    "\n",
    "- Идея - минимизировать отношение Релея\n",
    "- Используем градиентный спуск предобусловленный матрицей $B = A - \\sigma I$ \n",
    "\n",
    "$$ x_{k+1} = x_k - \\tau_k B^{-1}(Ax_k - R(x_k)x_k) $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy.sparse.linalg as spsplin\n",
    "import scipy.sparse as spsp\n",
    "\n",
    "def pinvit(A, x0, sigma, tau, num_iter, tol, inexact=True):\n",
    "    n = A.shape[0]\n",
    "    if inexact:\n",
    "        splu = spsplin.spilu(A - sigma * spsp.eye(n))\n",
    "    else:\n",
    "        splu = spsplin.splu(A - sigma * spsp.eye(n))\n",
    "    \n",
    "#     splu = spsplin.splu(spsp.eye(n))\n",
    "    x = x0.copy()\n",
    "    conv = [x]\n",
    "    for k in range(num_iter):\n",
    "        lam = (x @ A @ x) / (x @ x)\n",
    "        x = x - tau * splu.solve(A @ x - lam * x)\n",
    "        conv.append(x)\n",
    "        if np.linalg.norm(A @ x - lam * x) < tol:\n",
    "            break\n",
    "    return x, (x @ A @ x) / (x @ x), conv\n",
    "\n",
    "def invit(A, x0, tau, num_iter, tol):\n",
    "    n = A.shape[0]\n",
    "    x = x0.copy()\n",
    "    conv = [x]\n",
    "    for k in range(num_iter):\n",
    "        lam = (x @ A @ x) / (x @ x)\n",
    "        x = x - 2 * tau * (A @ x - lam * x) / (x @ x)\n",
    "        conv.append(x)\n",
    "        if np.linalg.norm(A @ x - lam * x) < tol:\n",
    "            break\n",
    "    return x, (x @ A @ x) / (x @ x), conv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.lines.Line2D at 0x7fa58e2c6550>"
      ]
     },
     "execution_count": 99,
     "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": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "n = 256\n",
    "ex = np.ones(n)\n",
    "lp1 = spsp.spdiags(np.vstack((ex,  -2*ex, ex)), [-1, 0, 1], n, n, 'csr')\n",
    "e = spsp.eye(n)\n",
    "A = spsp.kron(lp1, e) + spsp.kron(e, lp1)\n",
    "plt.spy(A, markersize=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.00029885]\n",
      "-0.000666678845291443\n",
      "-0.00029885332106977913\n",
      "-5.747387159537674\n"
     ]
    }
   ],
   "source": [
    "true_eigval, true_eigvec = spsplin.eigsh(A, which=\"SM\", k=1)\n",
    "print(true_eigval)\n",
    "x0 = np.random.randn(A.shape[0])\n",
    "# x0 = np.ones(A.shape[0])\n",
    "pinvit_eigvec, pinvit_eigval, conv_pinvit_inexact = pinvit(A, x0, 0, 1, 50, 1e-6, True)\n",
    "print(pinvit_eigval)\n",
    "pinvit_eigvec, pinvit_eigval, conv_pinvit_exact = pinvit(A, x0, 0, 1, 50, 1e-6, False)\n",
    "print(pinvit_eigval)\n",
    "invit_eigvec, invit_eigval, conv_invit = invit(A, x0, 40, 200, 1e-6)\n",
    "print(invit_eigval)\n",
    "# eigvals_all = np.linalg.eigvalsh(A.toarray())\n",
    "# np.sort(eigvals_all)[50:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7fa58dc99b50>"
      ]
     },
     "execution_count": 101,
     "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": [
    "plt.semilogy([np.linalg.norm(A @ x - (x @ A @ x) * x / (x @ x)) for x in conv_pinvit_exact], label=\"Exact\")\n",
    "plt.semilogy([np.linalg.norm(A @ x - (x @ A @ x) * x / (x @ x)) for x in conv_pinvit_inexact], label=\"Inexact\")\n",
    "# plt.semilogy([np.linalg.norm(A @ x - (x @ A @ x) * x / (x @ x)) for x in conv_invit], label=\"Invit\")\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "def shift_inverse_iteration_fast(A, x0, num_iter, shift, tol):\n",
    "    x = x0.copy()\n",
    "    conv = [x]\n",
    "    n = A.shape[0]\n",
    "    splu = spsplin.splu(A - shift * spsp.eye(n))\n",
    "    for i in range(num_iter):\n",
    "        x = splu.solve(x)\n",
    "        x = x / np.linalg.norm(x)\n",
    "        conv.append(x)\n",
    "        if np.linalg.norm(A @ x - x @ A @ x * x) < tol:\n",
    "            break\n",
    "    return x, conv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-0.0002988543363461039\n"
     ]
    }
   ],
   "source": [
    "shift = 0.\n",
    "num_iter = 100\n",
    "tol=1e-6\n",
    "middle_eigv, conv_m = shift_inverse_iteration_fast(A, x0, num_iter, shift, tol)\n",
    "print(middle_eigv @ A @ middle_eigv)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fa5a5ae8590>]"
      ]
     },
     "execution_count": 104,
     "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": [
    "plt.semilogy([np.linalg.norm(A @ x - (x @ A @ x) * x / (x @ x)) for x in conv_m])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7fa58c285a50>"
      ]
     },
     "execution_count": 106,
     "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": [
    "plt.semilogy([np.linalg.norm(A @ x - (x @ A @ x) * x) for x in conv_m], label=\"Shift-Invert\")\n",
    "plt.semilogy([np.linalg.norm(A @ x - (x @ A @ x) * x / (x @ x)) for x in conv_pinvit_exact], label=\"Exact\")\n",
    "plt.semilogy([np.linalg.norm(A @ x - (x @ A @ x) * x / (x @ x)) for x in conv_pinvit_inexact], label=\"Inexact\")\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 143,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Solving standard eigenvalue problem with preconditioning\n",
      "\n",
      "matrix size 65536\n",
      "block size 1\n",
      "\n",
      "No constraints\n",
      "\n",
      "\n",
      "iteration 0\n",
      "current block size: 1\n",
      "eigenvalue: [-3.9984872]\n",
      "residual norms: [1.99937588]\n",
      "iteration 1\n",
      "current block size: 1\n",
      "eigenvalue: [-0.02293889]\n",
      "residual norms: [0.1388527]\n",
      "iteration 2\n",
      "current block size: 1\n",
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    },
    {
     "name": "stdout",
     "output_type": "stream",
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      "final eigenvalue: [-0.00029886]\n",
      "final residual norms: [1.22043876e-05]\n"
     ]
    }
   ],
   "source": [
    "import scipy.sparse.linalg as spsplin\n",
    "\n",
    "X = np.random.randn(A.shape[0], 1)\n",
    "inc_lu = spsplin.spilu(A)\n",
    "prec = spsplin.LinearOperator(A.shape, matvec=lambda x: inc_lu.solve(x))\n",
    "eigenvalues, eigvect = spsplin.lobpcg(A, X, M=prec, tol=1e-6, largest=True, verbosityLevel=1, maxiter=900)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 144,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([-0.00029886]), (65536, 1))"
      ]
     },
     "execution_count": 144,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigenvalues, eigvect.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 145,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.2204387566041562e-05"
      ]
     },
     "execution_count": 145,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.norm(A @ eigvect.ravel() - eigenvalues[0] * eigvect.ravel())"
   ]
  }
 ],
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