{
  "cells": [
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "%matplotlib inline"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "\nOperators concatenation\n=======================\n\nThis example shows how to use 'stacking' operators such as\n:py:class:`pylops.VStack`, :py:class:`pylops.HStack`,\n:py:class:`pylops.Block`, :py:class:`pylops.BlockDiag`,\nand :py:class:`pylops.Kronecker`.\n\nThese operators allow for different combinations of multiple linear operators\nin a single operator. Such functionalities are used within PyLops as the basis\nfor the creation of complex operators as well as in the definition of various\ntypes of optimization problems with regularization or preconditioning.\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import numpy as np\nimport matplotlib.pyplot as plt\n\nimport pylops\n\nplt.close('all')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Let's start by defining two second derivatives :py:class:`pylops.SecondDerivative`\nthat we will be using in this example.\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "D2hop = pylops.SecondDerivative(11 * 21, dims=[11, 21], dir=1, dtype='float32')\nD2vop = pylops.SecondDerivative(11 * 21, dims=[11, 21], dir=0, dtype='float32')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Chaining of operators represents the simplest concatenation that\ncan be performed between two or more linear operators.\nThis can be easily achieved using the ``*`` symbol\n\n   .. math::\n      \\mathbf{D_{cat}}=  \\mathbf{D_v} \\mathbf{D_h}\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "Nv, Nh = 11, 21\nX = np.zeros((Nv, Nh))\nX[int(Nv/2), int(Nh/2)] = 1\n\nD2op = D2vop*D2hop\nY = np.reshape(D2op*X.flatten(), (Nv, Nh))\n\nfig, axs = plt.subplots(1, 2, figsize=(10, 3))\nfig.suptitle('Chain', fontsize=14,\n             fontweight='bold', y=0.95)\nim = axs[0].imshow(X, interpolation='nearest')\naxs[0].axis('tight')\naxs[0].set_title(r'$x$')\nplt.colorbar(im, ax=axs[0])\nim = axs[1].imshow(Y, interpolation='nearest')\naxs[1].axis('tight')\naxs[1].set_title(r'$y=(D_x+D_y) x$')\nplt.colorbar(im, ax=axs[1])\nplt.tight_layout()\nplt.subplots_adjust(top=0.8)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "We now want to *vertically stack* three operators\n\n   .. math::\n      \\mathbf{D_{Vstack}} =\n       \\begin{bmatrix}\n         \\mathbf{D_v}    \\\\\n         \\mathbf{D_h}\n       \\end{bmatrix}, \\qquad\n      \\mathbf{y} =\n       \\begin{bmatrix}\n         \\mathbf{D_v}\\mathbf{x}    \\\\\n         \\mathbf{D_h}\\mathbf{x}\n       \\end{bmatrix}\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "Nv, Nh = 11, 21\nX = np.zeros((Nv, Nh))\nX[int(Nv/2), int(Nh/2)] = 1\nDstack = pylops.VStack([D2vop, D2hop])\n\nY = np.reshape(Dstack * X.flatten(), (Nv * 2, Nh))\n\nfig, axs = plt.subplots(1, 2, figsize=(10, 3))\nfig.suptitle('Vertical stacking', fontsize=14,\n             fontweight='bold', y=0.95)\nim = axs[0].imshow(X, interpolation='nearest')\naxs[0].axis('tight')\naxs[0].set_title(r'$x$')\nplt.colorbar(im, ax=axs[0])\nim = axs[1].imshow(Y, interpolation='nearest')\naxs[1].axis('tight')\naxs[1].set_title(r'$y$')\nplt.colorbar(im, ax=axs[1])\nplt.tight_layout()\nplt.subplots_adjust(top=0.8)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Similarly we can now *horizontally stack* three operators\n\n   .. math::\n      \\mathbf{D_{Hstack}} =\n       \\begin{bmatrix}\n          \\mathbf{D_v}  & 0.5*\\mathbf{D_v} & -1*\\mathbf{D_h}\n       \\end{bmatrix}, \\qquad\n      \\mathbf{y} =\n       \\mathbf{D_v}\\mathbf{x}_1 + 0.5*\\mathbf{D_v}\\mathbf{x}_2 -\n       \\mathbf{D_h}\\mathbf{x}_3\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "Nv, Nh = 11, 21\nX = np.zeros((Nv*3, Nh))\nX[int(Nv/2), int(Nh/2)] = 1\nX[int(Nv/2) + Nv, int(Nh/2)] = 1\nX[int(Nv/2) + 2*Nv, int(Nh/2)] = 1\n\nHstackop = pylops.HStack([D2vop, 0.5 * D2vop, -1 * D2hop])\nY = np.reshape(Hstackop*X.flatten(), (Nv, Nh))\n\nfig, axs = plt.subplots(1, 2, figsize=(10, 3))\nfig.suptitle('Horizontal stacking', fontsize=14,\n             fontweight='bold', y=0.95)\nim = axs[0].imshow(X, interpolation='nearest')\naxs[0].axis('tight')\naxs[0].set_title(r'$x$')\nplt.colorbar(im, ax=axs[0])\nim = axs[1].imshow(Y, interpolation='nearest')\naxs[1].axis('tight')\naxs[1].set_title(r'$y$')\nplt.colorbar(im, ax=axs[1])\nplt.tight_layout()\nplt.subplots_adjust(top=0.8)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "We can even stack them both *horizontally* and *vertically* such that we\ncreate a *block* operator\n\n   .. math::\n      \\mathbf{D_{Block}} =\n       \\begin{bmatrix}\n          \\mathbf{D_v} & 0.5*\\mathbf{D_v} & -1*\\mathbf{D_h} \\\\\n          \\mathbf{D_h} & 2*\\mathbf{D_h}   & \\mathbf{D_v} \\\\\n       \\end{bmatrix}, \\qquad\n      \\mathbf{y} =\n       \\begin{bmatrix}\n          \\mathbf{D_v} \\mathbf{x_1} + 0.5*\\mathbf{D_v} \\mathbf{x_2} -\n          \\mathbf{D_h} \\mathbf{x_3} \\\\\n          \\mathbf{D_h} \\mathbf{x_1} + 2*\\mathbf{D_h} \\mathbf{x_2} +\n          \\mathbf{D_v} \\mathbf{x_3}\n       \\end{bmatrix}\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "Bop = pylops.Block([[D2vop, 0.5 * D2vop, -1 * D2hop],\n                    [D2hop, 2 * D2hop, D2vop]])\nY = np.reshape(Bop*X.flatten(), (2*Nv, Nh))\n\nfig, axs = plt.subplots(1, 2, figsize=(10, 3))\nfig.suptitle('Block', fontsize=14,\n             fontweight='bold', y=0.95)\nim = axs[0].imshow(X, interpolation='nearest')\naxs[0].axis('tight')\naxs[0].set_title(r'$x$')\nplt.colorbar(im, ax=axs[0])\nim = axs[1].imshow(Y, interpolation='nearest')\naxs[1].axis('tight')\naxs[1].set_title(r'$y$')\nplt.colorbar(im, ax=axs[1])\nplt.tight_layout()\nplt.subplots_adjust(top=0.8)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Finally we can use the *block-diagonal operator* to apply three operators\nto three different subset of the model and data\n\n   .. math::\n      \\mathbf{D_{BDiag}} =\n       \\begin{bmatrix}\n          \\mathbf{D_v}  & \\mathbf{0}       &  \\mathbf{0}  \\\\\n          \\mathbf{0}    & 0.5*\\mathbf{D_v} &  \\mathbf{0}  \\\\\n          \\mathbf{0}    & \\mathbf{0}       &  -\\mathbf{D_h}\n       \\end{bmatrix}, \\qquad\n      \\mathbf{y} =\n       \\begin{bmatrix}\n          \\mathbf{D_v}     \\mathbf{x_1}  \\\\\n          0.5*\\mathbf{D_v} \\mathbf{x_2}  \\\\\n          -\\mathbf{D_h}  \\mathbf{x_3}\n       \\end{bmatrix}\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "BD = pylops.BlockDiag([D2vop, 0.5 * D2vop, -1 * D2hop])\nY = np.reshape(BD*np.ndarray.flatten(X), (11*3, 21))\n\nfig, axs = plt.subplots(1, 2, figsize=(10, 3))\nfig.suptitle('Block-diagonal', fontsize=14,\n             fontweight='bold', y=0.95)\nim = axs[0].imshow(X, interpolation='nearest')\naxs[0].axis('tight')\naxs[0].set_title(r'$x$')\nplt.colorbar(im, ax=axs[0])\nim = axs[1].imshow(Y, interpolation='nearest')\naxs[1].axis('tight')\naxs[1].set_title(r'$y$')\nplt.colorbar(im, ax=axs[1])\nplt.tight_layout()\nplt.subplots_adjust(top=0.8)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Finally we use the *Kronecker operator* and replicate this example on\n`wiki <https://en.wikipedia.org/wiki/Kronecker_product>`_.\n\n   .. math::\n      \\begin{bmatrix}\n          1  & 2  \\\\\n          3  & 4 \\\\\n      \\end{bmatrix} \\otimes\n      \\begin{bmatrix}\n          0  & 5  \\\\\n          6  & 7 \\\\\n      \\end{bmatrix} =\n      \\begin{bmatrix}\n           0 &  5 &  0 & 10 \\\\\n           6 &  7 & 12 & 14 \\\\\n           0 & 15 &  0 & 20 \\\\\n          18 & 21 & 24 & 28 \\\\\n      \\end{bmatrix}\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "A = np.array([[1, 2], [3, 4]])\nB = np.array([[0, 5], [6, 7]])\nAB = np.kron(A, B)\n\nn1, m1 = A.shape\nn2, m2 = B.shape\n\nAop = pylops.MatrixMult(A)\nBop = pylops.MatrixMult(B)\n\nABop = pylops.Kronecker(Aop, Bop)\nx = np.ones(m1*m2)\n\ny = AB.dot(x)\nyop = ABop*x\nxinv = ABop / yop\n\nprint('AB = \\n', AB)\n\nprint('x = ', x)\nprint('y = ', y)\nprint('yop = ', yop)\nprint('xinv = ', x)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "We can also use :py:class:`pylops.Kronecker` to do something more\ninteresting. Any operator can in fact be applied on a single direction of a\nmulti-dimensional input array if combined with an :py:class:`pylops.Identity`\noperator via Kronecker product. We apply here the\n:py:class:`pylops.FirstDerivative` to the second dimension of the model.\n\nNote that for those operators whose implementation allows their application\nto a single axis via the ``dir`` parameter, using the Kronecker product\nwould lead to slower performance. Nevertheless, the Kronecker product allows\nany other operator to be applied to a single dimension.\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "Nv, Nh = 11, 21\n\nIop = pylops.Identity(Nv, dtype='float32')\nD2hop = pylops.FirstDerivative(Nh, dtype='float32')\n\nX = np.zeros((Nv, Nh))\nX[Nv//2, Nh//2] = 1\nD2hop = pylops.Kronecker(Iop, D2hop)\n\nY = D2hop*X.ravel()\nY = Y.reshape(Nv, Nh)\n\nfig, axs = plt.subplots(1, 2, figsize=(10, 3))\nfig.suptitle('Kronecker', fontsize=14,\n             fontweight='bold', y=0.95)\nim = axs[0].imshow(X, interpolation='nearest')\naxs[0].axis('tight')\naxs[0].set_title(r'$x$')\nplt.colorbar(im, ax=axs[0])\nim = axs[1].imshow(Y, interpolation='nearest')\naxs[1].axis('tight')\naxs[1].set_title(r'$y$')\nplt.colorbar(im, ax=axs[1])\nplt.tight_layout()\nplt.subplots_adjust(top=0.8)"
      ]
    }
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