simpeg/notebooks/Bound Constraint Problem.ipynb

{
 "metadata": {
  "name": ""
 },
 "nbformat": 3,
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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import SimPEG\n",
      "from SimPEG.mesh import TensorMesh\n",
      "from SimPEG.regularization import Regularization\n",
      "import SimPEG.inverse as inverse\n",
      "from SimPEG.inverse import Minimize\n",
      "import numpy as np\n",
      "import scipy.sparse as sp"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from SimPEG.forward.LinearProblem import example as LinExample\n",
      "\n",
      "prob, m_true = LinExample(1000)\n",
      "M = prob.mesh\n",
      "\n",
      "reg = Regularization(M)\n",
      "opt = inverse.InexactGaussNewton(maxIter=100,maxStep=0.2,debug=False,LSreduction=0.3,maxIterLS=50)\n",
      "opt.remember('f', ('norm_g', lambda M:np.linalg.norm(M.g)))\n",
      "inv = inverse.Inversion(prob,reg,opt,beta0=1e-4,maxIter=2,debug=False)\n",
      "m0 = np.zeros_like(m_true)\n",
      "\n",
      "mrec = inv.run(m0)\n",
      "\n",
      "plt.plot(M.vectorCCx, m_true, 'b-')\n",
      "plt.plot(M.vectorCCx, mrec, 'r-')\n",
      "\n",
      "figure()\n",
      "plot(opt.recall('f'))\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "====================== Inexact Gauss Newton ======================\n",
        "  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS  \n",
        "-----------------------------------------------------------------\n",
        "   0  1.00e-04  9.62e+02  0.00e+00  9.62e+02    1.02e+04      0  \n",
        "   1  1.00e-04  6.63e+02  7.73e+02  6.63e+02    8.47e+03      0  \n",
        "   2  1.00e-04  4.15e+02  4.39e+03  4.15e+02    6.63e+03      0  \n",
        "   3  1.00e-04  2.26e+02  6.85e+03  2.27e+02    4.80e+03      0  "
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "   4  1.00e-04  9.74e+01  1.07e+04  9.84e+01    3.02e+03      0  \n",
        "   5  1.00e-04  2.42e+01  1.64e+04  2.58e+01    1.35e+03      0  \n",
        "   6  1.00e-04  1.06e+00  2.34e+04  3.39e+00    9.21e+00      0  \n",
        "------------------------- STOP! -------------------------\n",
        "1 : |fc-fOld| = 2.2421e+01 <= tolF*(1+|f0|) = 9.6325e+01\n",
        "0 : |xc-x_last| = 2.1351e+00 <= tolX*(1+|x0|) = 1.0000e-01\n",
        "0 : |proj(x-g)-x|    = 9.2073e+00 <= tolG          = 1.0000e-01\n",
        "0 : |proj(x-g)-x|    = 9.2073e+00 <= 1e3*eps       = 1.0000e-02\n",
        "0 : maxIter   =     100    <= iter          =      6\n",
        "1 : phi_d  = 1.0595e+00 <= phi_d_target  = 2.0000e+01 \n",
        "------------------------- DONE! -------------------------\n",
        "-------------------------STOP!-------------------------\n",
        "0 : maxIter   =       2    <= iter          =      1\n",
        "1 : phi_d  = 1.0595e+00 <= phi_d_target  = 2.0000e+01 \n",
        "-------------------------DONE!-------------------------\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 3,
       "text": [
        "[<matplotlib.lines.Line2D at 0x10ebf9f10>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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b4kuJPJZZwx9dqbtn5kx5oZU7kp7BT7bElxJ5PJMP8GZlmbiwHeXmAqdPA08+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\nxXsAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10ecafa10>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10ecbec90>"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "FUN = SimPEG.tests.getQuadratic(sp.identity(2),np.array([-5,-5]))\n",
      "# FUN = SimPEG.tests.Rosenbrock\n",
      "\n",
      "f = FUN(np.array([1,2]))\n",
      "print f\n",
      "n,l,u = 50,-10,10\n",
      "I = np.zeros((n,n))\n",
      "X = np.linspace(l,u,n)\n",
      "for i, x in enumerate(X):\n",
      "    for j, y in enumerate(X):\n",
      "        f, g, H = FUN(np.array([x,y]))\n",
      "        I[i,j] = f\n",
      "\n",
      "colorbar(contourf(X,X, I.T))\n",
      "\n",
      "GN = inverse.GaussNewton()\n",
      "xopt = GN.minimize(FUN,np.array([0,0]))\n",
      "print xopt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "(-12.5, array([-4., -3.]), <2x2 sparse matrix of type '<type 'numpy.float64'>'\n",
        "\twith 2 stored elements (1 diagonals) in DIAgonal format>)\n",
        "=========== Gauss Newton ==========="
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "  #      f      |proj(x-g)-x|  LS  \n",
        "-----------------------------------\n",
        "   0  0.00e+00    7.07e+00      0  \n",
        "   1 -2.50e+01    0.00e+00      0  \n",
        "------------------------- STOP! -------------------------\n",
        "0 : |fc-fOld| = 2.5000e+01 <= tolF*(1+|f0|) = 1.0000e-01\n",
        "0 : |xc-x_last| = 7.0711e+00 <= tolX*(1+|x0|) = 1.0000e-01\n",
        "1 : |proj(x-g)-x|    = 0.0000e+00 <= tolG          = 1.0000e-01\n",
        "1 : |proj(x-g)-x|    = 0.0000e+00 <= 1e3*eps       = 1.0000e-02\n",
        "0 : maxIter   =      20    <= iter          =      1\n",
        "------------------------- DONE! -------------------------\n",
        "[ 5.  5.]\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10ebd9750>"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "opt = inverse.ProjectedGradient(maxIter=50,maxStep=np.inf, maxIterLS=20, debug=False)\n",
      "opt.remember('f')\n",
      "opt.lower = -2\n",
      "opt.upper = 2\n",
      "opt.minimize(FUN,np.array([0,0]))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "======================= Projected Gradient =======================\n",
        "  #      f      |proj(x-g)-x|  LS   itType   aSet    bSet  Comment\n",
        "------------------------------------------------------------------\n",
        "   0  0.00e+00    2.83e+00      0     SD      0       0           \n",
        "   1 -1.60e+01    0.00e+00      0     SD      2       2           \n",
        "------------------------- STOP! -------------------------\n",
        "0 : |fc-fOld| = 1.6000e+01 <= tolF*(1+|f0|) = 1.0000e-01\n",
        "0 : |xc-x_last| = 2.8284e+00 <= tolX*(1+|x0|) = 1.0000e-01\n",
        "1 : |proj(x-g)-x|    = 0.0000e+00 <= tolG          = 1.0000e-01\n",
        "1 : |proj(x-g)-x|    = 0.0000e+00 <= 1e3*eps       = 1.0000e-02\n",
        "0 : maxIter   =      50    <= iter          =      1\n",
        "1 : probSize  =      2   <= bindingSet      =      2\n",
        "------------------------- DONE! -------------------------\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "array([ 2.,  2.])"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "opt = inverse.ProjectedGradient(maxIter=50, maxStep=0.3,debug=False, maxIterLS=20, tolCG=1e-5, maxIterCG=1000)\n",
      "opt.lower, opt.upper = -0.4, 0.9\n",
      "opt.remember('f', 'xc', ('norm_g', lambda M: np.linalg.norm(M.g)), 'phi_d', 'phi_m')\n",
      "inv = inverse.Inversion(prob,reg,opt,beta0=1e-3,debug=False)\n",
      "inv.remember(('phi_d',lambda I:I.opt.recall('phi_d')),('phi_m',lambda I:I.opt.recall('phi_m')))\n",
      "m0 = np.zeros_like(m_true)\n",
      "mrecB = inv.run(m0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "====================================== Projected Gradient ======================================\n",
        "  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS   itType   aSet    bSet  Comment\n",
        "------------------------------------------------------------------------------------------------\n",
        "   0  1.00e-03  9.62e+02  0.00e+00  9.62e+02    2.14e+01      0     SD      0       0           \n",
        "   1  1.00e-03  9.53e+02  8.49e-03  9.53e+02    2.18e+01      9     SD      0       0           \n",
        "   2  1.00e-03  5.32e+02  1.42e+03  5.33e+02    2.20e+01      0    .CG.     0       0           "
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "   3  1.00e-03  2.32e+02  5.66e+03  2.38e+02    2.26e+01      0    .CG.     0       0           "
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "   4  1.00e-03  1.29e+02  8.85e+03  1.38e+02    2.30e+01      1    .CG.     0       0           "
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "   5  1.00e-03  1.10e+02  9.74e+03  1.20e+02    2.15e+01      3    .CG.     22      22          "
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "   6  1.00e-03  1.09e+02  9.74e+03  1.19e+02    2.36e+01     10     SD      22      0    Stop SD\n",
        "   7  1.00e-03  7.65e+01  1.33e+04  8.98e+01    2.21e+01      1    .CG.     85      75          "
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "   8  1.00e-03  5.11e+01  1.33e+04  6.44e+01    2.38e+01      9     SD      75      3           \n",
        "   9  1.00e-03  5.07e+01  1.33e+04  6.41e+01    2.11e+01     12     SD      4       4           \n",
        "  10  1.00e-03  5.06e+01  1.33e+04  6.40e+01    2.20e+01     12     SD      17      9    Stop SD\n",
        "  11  1.00e-03  5.04e+01  1.34e+04  6.37e+01    2.05e+01      8    .CG.     85      81          "
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "  12  1.00e-03  5.03e+01  1.34e+04  6.37e+01    1.65e+01     13     SD      81      26          \n",
        "  13  1.00e-03  5.03e+01  1.34e+04  6.37e+01    1.67e+01     15     SD      30      30          \n",
        "  14  1.00e-03  5.03e+01  1.34e+04  6.36e+01    1.42e+01     15     SD      84      84   Stop SD\n",
        "  15  1.00e-03  2.65e+01  2.26e+04  4.91e+01    2.52e+01      0    .CG.    190      50          "
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "  16  1.00e-03  1.51e+01  2.26e+04  3.77e+01    2.24e+01     10     SD      50      1           \n",
        "------------------------- STOP! -------------------------\n",
        "1 : |fc-fOld| = 1.1392e+01 <= tolF*(1+|f0|) = 9.6325e+01\n",
        "1 : |xc-x_last| = 3.4809e-03 <= tolX*(1+|x0|) = 1.0000e-01\n",
        "0 : |proj(x-g)-x|    = 2.2369e+01 <= tolG          = 1.0000e-01\n",
        "0 : |proj(x-g)-x|    = 2.2369e+01 <= 1e3*eps       = 1.0000e-02\n",
        "0 : maxIter   =      50    <= iter          =     16\n",
        "0 : probSize  =    1000   <= bindingSet      =      1\n",
        "1 : phi_d  = 1.5077e+01 <= phi_d_target  = 2.0000e+01 \n",
        "------------------------- DONE! -------------------------\n",
        "-------------------------STOP!-------------------------\n",
        "0 : maxIter   =      10    <= iter          =      1\n",
        "1 : phi_d  = 1.5077e+01 <= phi_d_target  = 2.0000e+01 \n",
        "-------------------------DONE!-------------------------\n"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# First set up the figure, the axis, and the plot element we want to animate\n",
      "fig = plt.figure()\n",
      "ax = plt.axes()\n",
      "ax.plot(M.vectorCCx, m_true, 'b-')\n",
      "txt = plt.text(0.8,0.9,'')\n",
      "line, = ax.plot([], [], 'r-', lw=1)\n",
      "\n",
      "def animate(i):\n",
      "    txt.set_text('iteration %d'%i)\n",
      "    line.set_data(M.vectorCCx, np.array(opt.recall('xc')[i]))\n",
      "\n",
      "SimPEG.utils.animate(fig, animate, frames=len(opt.recall('xc')))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
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        "\" type=\"video/mp4\">\n",
        " Your browser does not support the video tag.\n",
        "</video>"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 9,
       "text": [
        "<matplotlib.animation.FuncAnimation at 0x10ee35450>"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(opt.recall('phi_d'))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": [
        "[<matplotlib.lines.Line2D at 0x10eb01450>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x11030ec90>"
       ]
      }
     ],
     "prompt_number": 10
    }
   ],
   "metadata": {}
  }
 ]
}