04_havok_multiscale1.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import utils\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "from matplotlib import colors\n",
    "%matplotlib inline\n",
    "from IPython.display import set_matplotlib_formats\n",
    "set_matplotlib_formats('png', 'pdf')\n",
    "plt.rc('font', size=14)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "frequency_multipliers = 2**np.arange(3,8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## No spacing, large time step"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Frequency multiplier 8\n",
      "Frequency multiplier 16\n",
      "Frequency multiplier 32\n",
      "Frequency multiplier 64\n",
      "Frequency multiplier 128\n"
     ]
    }
   ],
   "source": [
    "downsample_errors = np.zeros(frequency_multipliers.shape)\n",
    "downsample_matrix_size = np.zeros((frequency_multipliers.size,2))\n",
    "\n",
    "for fm_idx,frequency_multiplier in enumerate(frequency_multipliers):\n",
    "    print(\"Frequency multiplier %d\" % frequency_multiplier)\n",
    "\n",
    "    mu1=5\n",
    "    mu2=5\n",
    "    c1 = 0\n",
    "    c2 = 0\n",
    "    tau1=.2\n",
    "    tau2=tau1*frequency_multiplier\n",
    "\n",
    "    x0=[0.0,2.0,2.0,0.0]\n",
    "    \n",
    "    fast_period_length = 11.45015*tau1\n",
    "    slow_period_length = fast_period_length*frequency_multiplier\n",
    "\n",
    "    num_periods_simulate = 10\n",
    "    sampling_rate_simulate = 2048\n",
    "    t_simulate = np.linspace(0, num_periods_simulate*slow_period_length,\n",
    "                    int(num_periods_simulate*sampling_rate_simulate*slow_period_length/fast_period_length))\n",
    "    dt_simulate = t_simulate[1]-t_simulate[0]\n",
    "\n",
    "    initial_samples = int(5*sampling_rate_simulate)\n",
    "    vdp_simulation = utils.simulate_coupled_vdp(dt_simulate, t_simulate.size+initial_samples, x0=x0,\n",
    "                                              mu1=mu1, mu2=mu2, c1=c1, c2=c2, tau1=tau1, tau2=tau2)[0]\n",
    "    vdp_solution = vdp_simulation[:,initial_samples:]\n",
    "    coupled_solution = vdp_solution[0:1] + vdp_solution[2:3]\n",
    "    \n",
    "    num_periods_subsample = 5\n",
    "    sampling_rate_subsample = 1024//frequency_multiplier\n",
    "\n",
    "    t_max_idx = int(num_periods_subsample*slow_period_length/dt_simulate)+2\n",
    "    spacing = sampling_rate_simulate//sampling_rate_subsample\n",
    "\n",
    "    t_sample = t_simulate[:t_max_idx:spacing]\n",
    "    dt_sample = t_sample[1] - t_sample[0]\n",
    "\n",
    "    sampled_data = coupled_solution[:,:t_max_idx:spacing]\n",
    "\n",
    "    time_delay=sampling_rate_subsample*frequency_multiplier\n",
    "    time_delay_spacing=1\n",
    "    rank=100\n",
    "    \n",
    "    sample_spacing=1\n",
    "    dt_scaling=1\n",
    "\n",
    "    dmd = utils.DMD(truncation='hard', threshold=rank, time_delay=time_delay, time_delay_spacing=time_delay_spacing)\n",
    "    dmd.fit(sampled_data, dt_sample, sample_spacing=sample_spacing, dt_scale=dt_scaling)\n",
    "    dmd_reconstruction = dmd.reconstruct(t_simulate, imaginary_evals=True)\n",
    "\n",
    "    error = np.mean((coupled_solution[0] - dmd_reconstruction[0])**2)\n",
    "    downsample_errors[fm_idx] = error\n",
    "    downsample_matrix_size[fm_idx] = utils.hankel_matrix(sampled_data, time_delay,\n",
    "                                                    spacing=time_delay_spacing)[:,dt_scaling::sample_spacing].shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## No spacing, small time step"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Frequency multiplier 8\n",
      "Frequency multiplier 16\n",
      "Frequency multiplier 32\n",
      "Frequency multiplier 64\n",
      "Frequency multiplier 128\n"
     ]
    }
   ],
   "source": [
    "nospacing_errors = np.zeros(frequency_multipliers.shape)\n",
    "nospacing_matrix_size = np.zeros((frequency_multipliers.size,2))\n",
    "\n",
    "for fm_idx,frequency_multiplier in enumerate(frequency_multipliers):\n",
    "    print(\"Frequency multiplier %d\" % frequency_multiplier)\n",
    "\n",
    "    mu1=5\n",
    "    mu2=5\n",
    "    c1 = 0\n",
    "    c2 = 0\n",
    "    tau1=.2\n",
    "    tau2=tau1*frequency_multiplier\n",
    "\n",
    "    x0=[0.0,2.0,2.0,0.0]\n",
    "    \n",
    "    fast_period_length = 11.45015*tau1\n",
    "    slow_period_length = fast_period_length*frequency_multiplier\n",
    "\n",
    "    num_periods_simulate = 10\n",
    "    sampling_rate_simulate = 2048\n",
    "    t_simulate = np.linspace(0, num_periods_simulate*slow_period_length,\n",
    "                    int(num_periods_simulate*sampling_rate_simulate*slow_period_length/fast_period_length))\n",
    "    dt_simulate = t_simulate[1]-t_simulate[0]\n",
    "\n",
    "    initial_samples = int(5*sampling_rate_simulate)\n",
    "    vdp_simulation = utils.simulate_coupled_vdp(dt_simulate, t_simulate.size+initial_samples, x0=x0,\n",
    "                                              mu1=mu1, mu2=mu2, c1=c1, c2=c2, tau1=tau1, tau2=tau2)[0]\n",
    "    vdp_solution = vdp_simulation[:,initial_samples:]\n",
    "    coupled_solution = vdp_solution[0:1] + vdp_solution[2:3]\n",
    "    \n",
    "    num_periods_subsample = 5\n",
    "    sampling_rate_subsample = 64\n",
    "\n",
    "    t_max_idx = int(num_periods_subsample*slow_period_length/dt_simulate)+2\n",
    "    spacing = sampling_rate_simulate//sampling_rate_subsample\n",
    "\n",
    "    t_sample = t_simulate[:t_max_idx:spacing]\n",
    "    dt_sample = t_sample[1] - t_sample[0]\n",
    "\n",
    "    sampled_data = coupled_solution[:,:t_max_idx:spacing]\n",
    "\n",
    "    time_delay=sampling_rate_subsample*frequency_multiplier\n",
    "    time_delay_spacing=1\n",
    "    rank=100\n",
    "    \n",
    "    sample_spacing=1\n",
    "    dt_scaling=1\n",
    "\n",
    "    dmd = utils.DMD(truncation='hard', threshold=rank, time_delay=time_delay, time_delay_spacing=time_delay_spacing)\n",
    "    dmd.fit(sampled_data, dt_sample, sample_spacing=sample_spacing, dt_scale=dt_scaling)\n",
    "    dmd_reconstruction = dmd.reconstruct(t_simulate, imaginary_evals=True)\n",
    "\n",
    "    error = np.mean((coupled_solution[0] - dmd_reconstruction[0])**2)\n",
    "    nospacing_errors[fm_idx] = error\n",
    "    nospacing_matrix_size[fm_idx] = utils.hankel_matrix(sampled_data, time_delay,\n",
    "                                                    spacing=time_delay_spacing)[:,dt_scaling::sample_spacing].shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Space out rows and columns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Frequency multiplier 8\n",
      "Frequency multiplier 16\n",
      "Frequency multiplier 32\n",
      "Frequency multiplier 64\n",
      "Frequency multiplier 128\n"
     ]
    }
   ],
   "source": [
    "fullspacing_errors = np.zeros(frequency_multipliers.shape)\n",
    "fullspacing_matrix_size = np.zeros((frequency_multipliers.size,2))\n",
    "\n",
    "for fm_idx,frequency_multiplier in enumerate(frequency_multipliers):\n",
    "    print(\"Frequency multiplier %d\" % frequency_multiplier)\n",
    "\n",
    "    mu1=5\n",
    "    mu2=5\n",
    "    c1 = 0\n",
    "    c2 = 0\n",
    "    tau1=.2\n",
    "    tau2=tau1*frequency_multiplier\n",
    "\n",
    "    x0=[0.0,2.0,2.0,0.0]\n",
    "    \n",
    "    fast_period_length = 11.45015*tau1\n",
    "    slow_period_length = fast_period_length*frequency_multiplier\n",
    "\n",
    "    num_periods_simulate = 10\n",
    "    sampling_rate_simulate = 2048\n",
    "    t_simulate = np.linspace(0, num_periods_simulate*slow_period_length,\n",
    "                    int(num_periods_simulate*sampling_rate_simulate*slow_period_length/fast_period_length))\n",
    "    dt_simulate = t_simulate[1]-t_simulate[0]\n",
    "\n",
    "    initial_samples = int(5*sampling_rate_simulate)\n",
    "    vdp_simulation = utils.simulate_coupled_vdp(dt_simulate, t_simulate.size+initial_samples, x0=x0,\n",
    "                                              mu1=mu1, mu2=mu2, c1=c1, c2=c2, tau1=tau1, tau2=tau2)[0]\n",
    "    vdp_solution = vdp_simulation[:,initial_samples:]\n",
    "    coupled_solution = vdp_solution[0:1] + vdp_solution[2:3]\n",
    "    \n",
    "    num_periods_subsample = 5\n",
    "    sampling_rate_subsample = 256\n",
    "\n",
    "    t_max_idx = int(num_periods_subsample*slow_period_length/dt_simulate)+2\n",
    "    spacing = sampling_rate_simulate//sampling_rate_subsample\n",
    "\n",
    "    t_sample = t_simulate[:t_max_idx:spacing]\n",
    "    dt_sample = t_sample[1] - t_sample[0]\n",
    "\n",
    "    sampled_data = coupled_solution[:,:t_max_idx:spacing]\n",
    "\n",
    "    time_delay=sampling_rate_subsample\n",
    "    time_delay_spacing=frequency_multiplier\n",
    "    rank=100\n",
    "    \n",
    "    sample_spacing=frequency_multiplier//frequency_multipliers[0]\n",
    "    dt_scaling=4\n",
    "\n",
    "    dmd = utils.DMD(truncation='hard', threshold=rank, time_delay=time_delay, time_delay_spacing=time_delay_spacing)\n",
    "    dmd.fit(sampled_data, dt_sample, sample_spacing=sample_spacing, dt_scale=dt_scaling)\n",
    "    dmd_reconstruction = dmd.reconstruct(t_simulate, imaginary_evals=True)\n",
    "\n",
    "    error = np.mean((coupled_solution[0,t_max_idx:] - dmd_reconstruction[0,t_max_idx:])**2)\n",
    "    fullspacing_errors[fm_idx] = error\n",
    "    fullspacing_matrix_size[fm_idx] = utils.hankel_matrix(sampled_data, time_delay,\n",
    "                                                    spacing=time_delay_spacing)[:,dt_scaling::sample_spacing].shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Plot results\n",
    "\n",
    "## Snapshot matrix size"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
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1U2FucyAvRmxhZ3MgMzIKL0ZvbnRCQm94IFsgLTEwMjEgLTQ2MyAxNzk0IDEyMzMgXSAv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\nMjUgMCBSID4+CnN0YXJ0eHJlZgo2MTU4CiUlRU9GCg==\n",
      "image/png": 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TU8NDDz3Eli1buPrqq+02zRAizFKTFjh58iT33Xcfmzdv5pJLLmHJkiXMnDnT7BfvYDhW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\nB2jAnHXQgPm0AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f2e4a374e10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(3,2.5))\n",
    "plt.loglog(frequency_multipliers, nospacing_matrix_size[:,0]*nospacing_matrix_size[:,1], '.-k', linewidth=2)\n",
    "plt.loglog(frequency_multipliers, downsample_matrix_size[:,0]*downsample_matrix_size[:,1], '.--k', linewidth=2)\n",
    "plt.loglog(frequency_multipliers, fullspacing_matrix_size[:,0]*fullspacing_matrix_size[:,1], '.:k', linewidth=2)\n",
    "plt.tight_layout()\n",
    "# plt.savefig('figures/05b_spacing_scaling.pdf', format='pdf', dpi=300)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/pdf": 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      "image/png": 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On7927RqDBg3ixIkTiIhJHD9nksdHBw8e5MqVKwDUrVuX5s2bc+XKFbp3705qairTpk3z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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f2de62fa9e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(3.1,2.5))\n",
    "plt.loglog(frequency_multipliers, nospacing_errors, '.-k')\n",
    "plt.loglog(frequency_multipliers, downsample_errors, '.--k')\n",
    "plt.loglog(frequency_multipliers, fullspacing_errors, '.:k')\n",
    "plt.tight_layout()\n",
    "# plt.savefig('figures/05b_spacing_errors.pdf', format='pdf', dpi=300)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "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": "3.6.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}