{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#Cahn-Hilliard with Primtive and Legendre Bases\n",
    "\n",
    "This example uses a Cahn-Hilliard model to compare two different bases representations to discretize the microstructure. One basis representaion uses the primitive (or hat) basis and the other uses Legendre polynomials. The example includes the background theory about using Legendre polynomials as a basis in MKS. The MKS with two different bases are compared with the standard spectral solution for the Cahn-Hilliard solution at both the calibration domain size and a scaled domain size. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###Cahn-Hilliard Equation\n",
    "\n",
    "The Cahn-Hilliard equation is used to simulate microstructure evolution during spinodial decomposition and has the following form,\n",
    "\n",
    "$$ \\dot{\\phi} = \\nabla^2 \\left( \\phi^3 - \\phi \\right) - \\gamma \\nabla^4 \\phi $$\n",
    "\n",
    "where $\\phi$ is a conserved ordered parameter and $\\sqrt{\\gamma}$ represents the width of the interface. In this example, the Cahn-Hilliard equation is solved using a semi-implicit spectral scheme with periodic boundary conditions, see  [Chang and Rutenberg](http://dx.doi.org/10.1103/PhysRevE.72.055701) for more details."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Basis Functions for the Microstructure Function and Influence Function\n",
    "\n",
    "In this example, we will explore the differences when using the\n",
    "Legendre polynomials as the basis function compared to the primitive\n",
    "(or hat) basis for the microstructure function and the influence coefficients.\n",
    "\n",
    "For more information about both of these basis please see the [theory section](THEORY.html)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%load_ext autoreload\n",
    "%autoreload 2\n",
    "\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##Modeling with MKS\n",
    "\n",
    "###Generating Calibration Datasets\n",
    "\n",
    "Because the microstructure is a continuous field that can have a range of values and changes over time, the first order influence coefficients cannot be calibrated with delta microstructures. Instead a large number of simulations with random initial conditions will be used to calibrate the first order influence coefficients using linear regression. Let's show how this is done.\n",
    "\n",
    "The function `make_cahnHilliard` from `pymks.datasets` provides a nice interface to generate calibration datasets for the influence coefficients. The funcion `make_cahnHilliard` requires the number of calibration samples given by `n_samples` and the size and shape of the domain given by `size`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import pymks\n",
    "from pymks.datasets import make_cahn_hilliard\n",
    "\n",
    "length = 41\n",
    "n_samples = 400\n",
    "dt = 1e-2\n",
    "np.random.seed(101)\n",
    "size=(length, length)\n",
    "X, y = make_cahn_hilliard(n_samples=n_samples, size=size, dt=dt)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function `make_cahnHilliard` has generated `n_samples` number of random microstructures, `X`, and returned the same microstructures after they have evolved for one time step given by `y`. Let's take a look at one of them."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "QmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa6db1e1b50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from pymks.tools import draw_concentrations\n",
    "\n",
    "draw_concentrations((X[0], y[0]),('Calibration Input', 'Calibration Output'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calibrate Influence Coefficients\n",
    " \n",
    "In this example, we compare the difference between using the primitive (or hat) basis and the Legendre polynomial basis to represent the microstructure function. As mentioned above, the microstructures (concentration fields) are not discrete phases. This leaves the number of local states in local state space `n_states` as a free hyper parameter. In the next section we look to see what a practical number of local states for bases would be. \n",
    " \n",
    "#### Optimizing the Number of Local States\n",
    " \n",
    "Below, we compare the difference in performance as we vary the local state when we choose the primitive basis and the Legendre polynomial basis.\n",
    "\n",
    "The `(X, y)` sample data is split into training and test data. The code then optimizes `n_states` between `2` and `11` and the two `basis` with the `parameters_to_tune` variable. The `GridSearchCV` takes an `MKSLocalizationModel` instance, a `scoring` function (figure of merit) and the `parameters_to_tune` and then finds the optimal parameters with a grid search."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from pymks.bases import PrimitiveBasis\n",
    "from sklearn.grid_search import GridSearchCV\n",
    "from sklearn import metrics\n",
    "mse = metrics.mean_squared_error\n",
    "from pymks.bases import LegendreBasis\n",
    "from pymks import MKSLocalizationModel\n",
    "from sklearn.cross_validation import train_test_split\n",
    "\n",
    "train_split_shape = (X.shape[0],) + (np.prod(X.shape[1:]),)\n",
    "\n",
    "X_train, X_test, y_train, y_test = train_test_split(X.reshape(train_split_shape),\n",
    "                                                    y.reshape(train_split_shape),\n",
    "                                                    test_size=0.5, random_state=3)\n",
    "\n",
    "prim_basis = PrimitiveBasis(2, [-1, 1])\n",
    "leg_basis = LegendreBasis(2, [-1, 1])\n",
    "\n",
    "params_to_tune = {'n_states': np.arange(2, 11),\n",
    "                 'basis': [prim_basis, leg_basis]}\n",
    "Model = MKSLocalizationModel(prim_basis)\n",
    "scoring = metrics.make_scorer(lambda a, b: -mse(a, b))\n",
    "fit_params = {'size': size}\n",
    "gs = GridSearchCV(Model, params_to_tune, cv=5, fit_params=fit_params, n_jobs=3).fit(X_train, y_train)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The optimal parameters are the `LegendreBasis` with only 4 local states. More terms don't improve the R-squared value."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "MKSLocalizationModel(basis=<pymks.bases.legendre.LegendreBasis object at 0x7fa6f49e4210>,\n",
      "           n_states=4)\n",
      "1.0\n"
     ]
    }
   ],
   "source": [
    "print(gs.best_estimator_)\n",
    "print(gs.score(X_test, y_test))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "CIcjCYwQQgghHM7/A+mNOGjuV1pRAAAAAElFTkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa6cc79ced0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from pymks.tools import draw_gridscores\n",
    "\n",
    "lgs = [x for x in gs.grid_scores_ \\\n",
    "       if type(x.parameters['basis']) is type(leg_basis)]\n",
    "cgs = [x for x in gs.grid_scores_ \\\n",
    "       if type(x.parameters['basis']) is type(prim_basis)]\n",
    "\n",
    "draw_gridscores([lgs, cgs], 'n_states', data_labels=['Legendre', 'Primitve'],\n",
    "                colors=['#f46d43', '#1a9641'], score_label='R-Squared', \n",
    "                param_label = 'L - Total Number of Local States')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you can see the `LegendreBasis` converges faster than the `PrimitiveBasis`. In order to further compare performance between the two models, lets select 4 local states for both bases."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Comparing the Bases for `n_states=4`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "prim_basis = PrimitiveBasis(n_states=4, domain=[-1, 1])\n",
    "prim_model = MKSLocalizationModel(basis=prim_basis)\n",
    "prim_model.fit(X, y)\n",
    "\n",
    "leg_basis = LegendreBasis(4, [-1, 1])\n",
    "leg_model = MKSLocalizationModel(basis=leg_basis)\n",
    "leg_model.fit(X, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's look at the influence coefficients for both bases.\n",
    "\n",
    "First the `PrimitiveBasis` influence coefficients"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "AAAAACOIAA4AAAAAAAAYRgRwAAAAAAAAwDAigAMAAAAAAACGEQEcAAAAAAAAMIwI4AAAAAAAAIBh\n",
       "RAAHAAAAAAAADCMCOAAAAAAAAGAYEcABAAAAAAAAw+j/AwXVFYf0hEIFAAAAAElFTkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa6f56ad1d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from pymks.tools import draw_coeff\n",
    "\n",
    "draw_coeff(prim_model.coeff)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now for the `LegendreBasis` influence coefficients."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa6cc574050>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "draw_coeff(leg_model.coeff)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's do some simulations with both sets of coefficients and compare the results."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###Predict Microstructure Evolution\n",
    "\n",
    "In order to compare the difference between the two bases, we need to have the Cahn-Hilliard simulation and the two MKS models start with the same initial concentration `phi0` and evolve in time. In order to do the Cahn-Hilliard simulation we need an instance of the class `CahnHilliardSimulation`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from pymks.datasets.cahn_hilliard_simulation import CahnHilliardSimulation\n",
    "np.random.seed(66)\n",
    "\n",
    "phi0 = np.random.normal(0, 1e-9, ((1,) + size))\n",
    "ch_sim = CahnHilliardSimulation(dt=dt)\n",
    "phi_sim = phi0.copy()\n",
    "phi_prim = phi0.copy()\n",
    "phi_legendre = phi0.copy()\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's look at the inital concentration field."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "TkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa6cc5dc5d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "draw_concentrations([phi0[0]], ['Initial Concentration'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to move forward in time, we need to feed the concentration back into the Cahn-Hilliard simulation and the MKS models."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "time_steps = 50\n",
    "\n",
    "for steps in range(time_steps):\n",
    "    ch_sim.run(phi_sim)\n",
    "    phi_sim = ch_sim.response\n",
    "    phi_prim = prim_model.predict(phi_prim)\n",
    "    phi_legendre = leg_model.predict(phi_legendre)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's take a look at the concentration fields."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "AAAAgCOHwSoAAAAA4Mj5D0CIHUFQj2GpAAAAAElFTkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa6cc79ced0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from pymks.tools import draw_concentrations\n",
    "\n",
    "draw_concentrations((phi_sim[0], phi_prim[0], phi_legendre[0]),\n",
    "                    ('Simulation', 'Primative', 'Legendre'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By just looking at the three microstructures is it difficult to see any differences. Below, we plot the difference between the two MKS models and the simulation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa6cc330410>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Primative mse = 5.28702717916e-23\n",
      "Legendre mse = 4.35706317904e-28\n"
     ]
    }
   ],
   "source": [
    "from sklearn import metrics\n",
    "mse = metrics.mean_squared_error\n",
    "from pymks.tools import draw_differences\n",
    "\n",
    "draw_differences([(phi_sim[0] - phi_prim[0]), (phi_sim[0] - phi_legendre[0])],\n",
    "                 ['Simulaiton - Prmitive', 'Simulation - Legendre'])\n",
    "\n",
    "print 'Primative mse =', mse(phi_sim[0], phi_prim[0])\n",
    "print 'Legendre mse =', mse(phi_sim[0], phi_legendre[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `LegendreBasis` basis clearly out performs the `PrimitiveBasis` for the same value of `n_states`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##Resizing the Coefficients to use on Larger Systems \n",
    "\n",
    "Below we compare the bases after the coefficients are resized."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "big_length = 3 * length\n",
    "big_size = (big_length, big_length)\n",
    "prim_model.resize_coeff(big_size)\n",
    "leg_model.resize_coeff(big_size)\n",
    "\n",
    "phi0 = np.random.normal(0, 1e-9, (1,) + big_size)\n",
    "phi_sim = phi0.copy()\n",
    "phi_prim = phi0.copy()\n",
    "phi_legendre = phi0.copy()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's take a look at the initial large concentration field."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "DvG/r6Xjx7o92FUAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa6cc338910>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "draw_concentrations([phi0[0]], ['Initial Concentration'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's look at the resized coefficients.\n",
    "\n",
    "First the influence coefficients from the `PrimitiveBasis`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "AAAAAACcQ/8f9yIfpGxGr8MAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa6cc5e2d90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "draw_coeff(prim_model.coeff)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now the influence coefficients from the `LegendreBases`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa6cc62a490>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "draw_coeff(leg_model.coeff)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Once again we are going to march forward in time by feeding the concentration fields back into the Cahn-Hilliard simulation and the MKS models. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "for steps in range(time_steps):\n",
    "    ch_sim.run(phi_sim)\n",
    "    phi_sim = ch_sim.response\n",
    "    phi_prim = prim_model.predict(phi_prim)\n",
    "    phi_legendre = leg_model.predict(phi_legendre)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa6cc655610>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "draw_concentrations((phi_sim[0], phi_prim[0], phi_legendre[0]), ('Simulation', 'Primiative', 'Legendre'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Both the MKS models seem to predict the concentration faily well. However, the Legendre polynomial basis looks to be better. Again let's look at the difference between the simulation and the MKS models."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa6cc68d7d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Primative mse = 4.43202082745e-23\n",
      "Legendre mse = 4.45272856882e-28\n"
     ]
    }
   ],
   "source": [
    "draw_differences([(phi_sim[0] - phi_prim[0]), (phi_sim[0] - phi_legendre[0])], \n",
    "                 ['Simulaiton - Primiative','Simulation - Legendre'])\n",
    "\n",
    "print 'Primative mse =', mse(phi_sim[0], phi_prim[0])\n",
    "print 'Legendre mse =', mse(phi_sim[0], phi_legendre[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With the resized influence coefficients, the `LegendreBasis` outperforms the `PrimitiveBasis` for the same value of `n_states`. The value of `n_states` does not necessarily guarantee a fair comparison between the two basis in terms of floating point calculations and memory used."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}