Computational_Seismology_Igel/se_numerical_integration_GLL.ipynb

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    "<div style='background-image: url(\"../../share/images/header.svg\") ; padding: 0px ; background-size: cover ; border-radius: 5px ; height: 250px'>\n",
    "    <div style=\"float: right ; margin: 50px ; padding: 20px ; background: rgba(255 , 255 , 255 , 0.7) ; width: 50% ; height: 150px\">\n",
    "        <div style=\"position: relative ; top: 50% ; transform: translatey(-50%)\">\n",
    "            <div style=\"font-size: xx-large ; font-weight: 900 ; color: rgba(0 , 0 , 0 , 0.8) ; line-height: 100%\">Computational Seismology</div>\n",
    "            <div style=\"font-size: large ; padding-top: 20px ; color: rgba(0 , 0 , 0 , 0.5)\">Numerical Integration - The Gauss-Lobatto-Legendre approach</div>\n",
    "        </div>\n",
    "    </div>\n",
    "</div>"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"width:20%;float:right;padding-left:50px\">\n",
    "<img src=../../share/images/book.jpg>\n",
    "<span style=\"font-size:smaller\">\n",
    "</span>\n",
    "</p>\n",
    "\n",
    "\n",
    "---\n",
    "\n",
    "This notebook is part of the supplementary material\n",
    "to [Computational Seismology: A Practical Introduction](https://global.oup.com/academic/product/computational-seismology-9780198717416?cc=de&lang=en&#),\n",
    "Oxford University Press, 2016.\n",
    "\n",
    "\n",
    "##### Authors:\n",
    "* Heiner Igel ([@heinerigel](https://github.com/heinerigel))\n",
    "* Stephanie Wollherr ([@swollherr](https://github.com/swollherr))\n",
    "* Florian Wölfl ([@flo-woelfl](https://github.com/flo-woelfl))\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following notebook presents a basic integration scheme that we're going to use in the Spectral Element Code as well as in the Discontinuous Galerkin Code to calculate the entries of the mass and stiffness matrices."
   ]
  },
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   "source": [
    "**Fundamental principal:**<br>\n",
    "Replace the function $f(x)$ that we want to integrate by a polynomial approximation that can be integrated analytically.\n",
    "\n",
    "As interpolating functions we use the Lagrange polynomials $l_i$ and obtain the following integration scheme for an arbitrary function $f(x)$ defined on the interval $[-1,1]$ :\n",
    "\\begin{eqnarray*} \\int_{-1}^1 f(x) \\ dx \\approx \\int _{-1}^1 P_N(x) dx = \\sum_{i=1}^{N+1} w_i f(x_i) \\end{eqnarray*}\n",
    "with\n",
    "\\begin{eqnarray*}\n",
    "    P_N(x)= \\sum_{i=1}^{N+1} f(x_i) \\ l_i^{(N)}(x).\n",
    "\\end{eqnarray*}\n",
    "As collocation points we use the Gauss-Lobatto-Legendre points $x_i$ and the corresponding weights that are needed to evaluate the integral are calculated as follows:\n",
    "\\begin{eqnarray*} w_i= \\int_{-1}^1 l_i^{(N)}(x) \\ dx \\end{eqnarray*}."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercises\n",
    "We want to investigate the performance of\n",
    "the numerical integration scheme. You can use the `gll()` routine to\n",
    "obtain the differentiation weights $w_i$ for an\n",
    "arbitrary function f(x) and the relevant integration points $x_i$.\n",
    "\n",
    "### 1. Numerical integration of an arbritrary function:\n",
    "Define a function $f(x)$\n",
    "of your choice and calculate analytically the\n",
    "integral $\\int f(x) \\ dx$ for the interval $[−1, 1]$. Perform the integration numerically and compare the results.\n",
    "\n",
    "### 2. The order of integration\n",
    "Modify the function and\n",
    "the order of the numerical integration. Discuss the results.\n",
    "\n",
    "### Have fun!"
   ]
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    "# This is a configuration step for the exercise. Please run it before the simulation code!\n",
    "import numpy as np\n",
    "import matplotlib\n",
    "# Show Plot in The Notebook\n",
    "matplotlib.use(\"nbagg\")\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "from gll import gll\n",
    "from lagrange2 import lagrange2\n",
    "\n",
    "# Prettier plots.\n",
    "plt.style.use('ggplot')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
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   "outputs": [
    {
     "name": "stdout",
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     "text": [
      "Solution of the analytical integral: 0\n",
      "Solution of the numerical integral: 1.22465e-17\n"
     ]
    },
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       "\n",
       "        var msg = JSON.parse(evt.data);\n",
       "        var msg_type = msg['type'];\n",
       "\n",
       "        // Call the  \"handle_{type}\" callback, which takes\n",
       "        // the figure and JSON message as its only arguments.\n",
       "        try {\n",
       "            var callback = fig[\"handle_\" + msg_type];\n",
       "        } catch (e) {\n",
       "            console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        if (callback) {\n",
       "            try {\n",
       "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
       "                callback(fig, msg);\n",
       "            } catch (e) {\n",
       "                console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n",
       "            }\n",
       "        }\n",
       "    };\n",
       "}\n",
       "\n",
       "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
       "mpl.findpos = function(e) {\n",
       "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
       "    var targ;\n",
       "    if (!e)\n",
       "        e = window.event;\n",
       "    if (e.target)\n",
       "        targ = e.target;\n",
       "    else if (e.srcElement)\n",
       "        targ = e.srcElement;\n",
       "    if (targ.nodeType == 3) // defeat Safari bug\n",
       "        targ = targ.parentNode;\n",
       "\n",
       "    // jQuery normalizes the pageX and pageY\n",
       "    // pageX,Y are the mouse positions relative to the document\n",
       "    // offset() returns the position of the element relative to the document\n",
       "    var x = e.pageX - $(targ).offset().left;\n",
       "    var y = e.pageY - $(targ).offset().top;\n",
       "\n",
       "    return {\"x\": x, \"y\": y};\n",
       "};\n",
       "\n",
       "/*\n",
       " * return a copy of an object with only non-object keys\n",
       " * we need this to avoid circular references\n",
       " * http://stackoverflow.com/a/24161582/3208463\n",
       " */\n",
       "function simpleKeys (original) {\n",
       "  return Object.keys(original).reduce(function (obj, key) {\n",
       "    if (typeof original[key] !== 'object')\n",
       "        obj[key] = original[key]\n",
       "    return obj;\n",
       "  }, {});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.mouse_event = function(event, name) {\n",
       "    var canvas_pos = mpl.findpos(event)\n",
       "\n",
       "    if (name === 'button_press')\n",
       "    {\n",
       "        this.canvas.focus();\n",
       "        this.canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    var x = canvas_pos.x * mpl.ratio;\n",
       "    var y = canvas_pos.y * mpl.ratio;\n",
       "\n",
       "    this.send_message(name, {x: x, y: y, button: event.button,\n",
       "                             step: event.step,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "\n",
       "    /* This prevents the web browser from automatically changing to\n",
       "     * the text insertion cursor when the button is pressed.  We want\n",
       "     * to control all of the cursor setting manually through the\n",
       "     * 'cursor' event from matplotlib */\n",
       "    event.preventDefault();\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    // Handle any extra behaviour associated with a key event\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.key_event = function(event, name) {\n",
       "\n",
       "    // Prevent repeat events\n",
       "    if (name == 'key_press')\n",
       "    {\n",
       "        if (event.which === this._key)\n",
       "            return;\n",
       "        else\n",
       "            this._key = event.which;\n",
       "    }\n",
       "    if (name == 'key_release')\n",
       "        this._key = null;\n",
       "\n",
       "    var value = '';\n",
       "    if (event.ctrlKey && event.which != 17)\n",
       "        value += \"ctrl+\";\n",
       "    if (event.altKey && event.which != 18)\n",
       "        value += \"alt+\";\n",
       "    if (event.shiftKey && event.which != 16)\n",
       "        value += \"shift+\";\n",
       "\n",
       "    value += 'k';\n",
       "    value += event.which.toString();\n",
       "\n",
       "    this._key_event_extra(event, name);\n",
       "\n",
       "    this.send_message(name, {key: value,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n",
       "    if (name == 'download') {\n",
       "        this.handle_save(this, null);\n",
       "    } else {\n",
       "        this.send_message(\"toolbar_button\", {name: name});\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n",
       "    this.message.textContent = tooltip;\n",
       "};\n",
       "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
       "\n",
       "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n",
       "\n",
       "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n",
       "    // Create a \"websocket\"-like object which calls the given IPython comm\n",
       "    // object with the appropriate methods. Currently this is a non binary\n",
       "    // socket, so there is still some room for performance tuning.\n",
       "    var ws = {};\n",
       "\n",
       "    ws.close = function() {\n",
       "        comm.close()\n",
       "    };\n",
       "    ws.send = function(m) {\n",
       "        //console.log('sending', m);\n",
       "        comm.send(m);\n",
       "    };\n",
       "    // Register the callback with on_msg.\n",
       "    comm.on_msg(function(msg) {\n",
       "        //console.log('receiving', msg['content']['data'], msg);\n",
       "        // Pass the mpl event to the overridden (by mpl) onmessage function.\n",
       "        ws.onmessage(msg['content']['data'])\n",
       "    });\n",
       "    return ws;\n",
       "}\n",
       "\n",
       "mpl.mpl_figure_comm = function(comm, msg) {\n",
       "    // This is the function which gets called when the mpl process\n",
       "    // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
       "\n",
       "    var id = msg.content.data.id;\n",
       "    // Get hold of the div created by the display call when the Comm\n",
       "    // socket was opened in Python.\n",
       "    var element = $(\"#\" + id);\n",
       "    var ws_proxy = comm_websocket_adapter(comm)\n",
       "\n",
       "    function ondownload(figure, format) {\n",
       "        window.open(figure.imageObj.src);\n",
       "    }\n",
       "\n",
       "    var fig = new mpl.figure(id, ws_proxy,\n",
       "                           ondownload,\n",
       "                           element.get(0));\n",
       "\n",
       "    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
       "    // web socket which is closed, not our websocket->open comm proxy.\n",
       "    ws_proxy.onopen();\n",
       "\n",
       "    fig.parent_element = element.get(0);\n",
       "    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
       "    if (!fig.cell_info) {\n",
       "        console.error(\"Failed to find cell for figure\", id, fig);\n",
       "        return;\n",
       "    }\n",
       "\n",
       "    var output_index = fig.cell_info[2]\n",
       "    var cell = fig.cell_info[0];\n",
       "\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_close = function(fig, msg) {\n",
       "    var width = fig.canvas.width/mpl.ratio\n",
       "    fig.root.unbind('remove')\n",
       "\n",
       "    // Update the output cell to use the data from the current canvas.\n",
       "    fig.push_to_output();\n",
       "    var dataURL = fig.canvas.toDataURL();\n",
       "    // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
       "    // the notebook keyboard shortcuts fail.\n",
       "    IPython.keyboard_manager.enable()\n",
       "    $(fig.parent_element).html('<img src=\"' + dataURL + '\" width=\"' + width + '\">');\n",
       "    fig.close_ws(fig, msg);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.close_ws = function(fig, msg){\n",
       "    fig.send_message('closing', msg);\n",
       "    // fig.ws.close()\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n",
       "    // Turn the data on the canvas into data in the output cell.\n",
       "    var width = this.canvas.width/mpl.ratio\n",
       "    var dataURL = this.canvas.toDataURL();\n",
       "    this.cell_info[1]['text/html'] = '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Tell IPython that the notebook contents must change.\n",
       "    IPython.notebook.set_dirty(true);\n",
       "    this.send_message(\"ack\", {});\n",
       "    var fig = this;\n",
       "    // Wait a second, then push the new image to the DOM so\n",
       "    // that it is saved nicely (might be nice to debounce this).\n",
       "    setTimeout(function () { fig.push_to_output() }, 1000);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items){\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) { continue; };\n",
       "\n",
       "        var button = $('<button class=\"btn btn-default\" href=\"#\" title=\"' + name + '\"><i class=\"fa ' + image + ' fa-lg\"></i></button>');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    // Add the status bar.\n",
       "    var status_bar = $('<span class=\"mpl-message\" style=\"text-align:right; float: right;\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "\n",
       "    // Add the close button to the window.\n",
       "    var buttongrp = $('<div class=\"btn-group inline pull-right\"></div>');\n",
       "    var button = $('<button class=\"btn btn-mini btn-primary\" href=\"#\" title=\"Stop Interaction\"><i class=\"fa fa-power-off icon-remove icon-large\"></i></button>');\n",
       "    button.click(function (evt) { fig.handle_close(fig, {}); } );\n",
       "    button.mouseover('Stop Interaction', toolbar_mouse_event);\n",
       "    buttongrp.append(button);\n",
       "    var titlebar = this.root.find($('.ui-dialog-titlebar'));\n",
       "    titlebar.prepend(buttongrp);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(el){\n",
       "    var fig = this\n",
       "    el.on(\"remove\", function(){\n",
       "\tfig.close_ws(fig, {});\n",
       "    });\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(el){\n",
       "    // this is important to make the div 'focusable\n",
       "    el.attr('tabindex', 0)\n",
       "    // reach out to IPython and tell the keyboard manager to turn it's self\n",
       "    // off when our div gets focus\n",
       "\n",
       "    // location in version 3\n",
       "    if (IPython.notebook.keyboard_manager) {\n",
       "        IPython.notebook.keyboard_manager.register_events(el);\n",
       "    }\n",
       "    else {\n",
       "        // location in version 2\n",
       "        IPython.keyboard_manager.register_events(el);\n",
       "    }\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    var manager = IPython.notebook.keyboard_manager;\n",
       "    if (!manager)\n",
       "        manager = IPython.keyboard_manager;\n",
       "\n",
       "    // Check for shift+enter\n",
       "    if (event.shiftKey && event.which == 13) {\n",
       "        this.canvas_div.blur();\n",
       "        event.shiftKey = false;\n",
       "        // Send a \"J\" for go to next cell\n",
       "        event.which = 74;\n",
       "        event.keyCode = 74;\n",
       "        manager.command_mode();\n",
       "        manager.handle_keydown(event);\n",
       "    }\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    fig.ondownload(fig, null);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.find_output_cell = function(html_output) {\n",
       "    // Return the cell and output element which can be found *uniquely* in the notebook.\n",
       "    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
       "    // IPython event is triggered only after the cells have been serialised, which for\n",
       "    // our purposes (turning an active figure into a static one), is too late.\n",
       "    var cells = IPython.notebook.get_cells();\n",
       "    var ncells = cells.length;\n",
       "    for (var i=0; i<ncells; i++) {\n",
       "        var cell = cells[i];\n",
       "        if (cell.cell_type === 'code'){\n",
       "            for (var j=0; j<cell.output_area.outputs.length; j++) {\n",
       "                var data = cell.output_area.outputs[j];\n",
       "                if (data.data) {\n",
       "                    // IPython >= 3 moved mimebundle to data attribute of output\n",
       "                    data = data.data;\n",
       "                }\n",
       "                if (data['text/html'] == html_output) {\n",
       "                    return [cell, data, j];\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "// Register the function which deals with the matplotlib target/channel.\n",
       "// The kernel may be null if the page has been refreshed.\n",
       "if (IPython.notebook.kernel != null) {\n",
       "    IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n",
       "}\n"
      ],
      "text/plain": [
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     "metadata": {},
     "output_type": "display_data"
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    {
     "data": {
      "text/html": [
       "<div id='c5777411-ea6a-46bb-8c1b-6b1008916b35'></div>"
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      "text/plain": [
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     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "# Exercise for Gauss integration\n",
    "\n",
    "n = 1000\n",
    "x = np.linspace(-1, 1, n)\n",
    "\n",
    "# MODIFY f and intf to test different functions!\n",
    "f = np.sin(x * np.pi)\n",
    "\n",
    "# Analytical value of the DEFINITE integral from -1 to 1\n",
    "intf = 1.0 / np.pi * (-np.cos(1.0 * np.pi) + np.cos(-1.0 * np.pi))\n",
    "\n",
    "# Choose order\n",
    "N = 4\n",
    "# Uncomment for interactivity.\n",
    "# N =int(input('Give order of integration: '))\n",
    "\n",
    "# Get integration points and weights from the gll routine\n",
    "xi, w = gll(N)\n",
    "\n",
    "# Initialize function at points xi\n",
    "fi = np.interp(xi, x, f)\n",
    "\n",
    "# Evaluate integral\n",
    "intfn = 0\n",
    "for i in range(len(w)):\n",
    "    intfn = intfn + w[i] * fi[i]\n",
    "\n",
    "# Calculate Lagrange Interpolant for plotting purposes.\n",
    "lp = np.zeros((N + 1, len(x)))\n",
    "for i in range(0, len(x)):\n",
    "    for j in range(-1, N):\n",
    "        lp[j + 1, i] = lagrange2(N, j, x[i], xi)\n",
    "s = np.zeros_like(x)\n",
    "for j in range(0, N + 1):\n",
    "    s = s + lp[j, :] * fi[j]\n",
    "\n",
    "print('Solution of the analytical integral: %g' % intf)\n",
    "print('Solution of the numerical integral: %g' % intfn)\n",
    "\n",
    "# Plot results.\n",
    "plt.figure(figsize=(10, 5))\n",
    "plt.plot(x, f, 'k-', label='Analytical Function')\n",
    "\n",
    "plt.plot(xi, fi, 'bs', label='GLL points')\n",
    "plt.plot(x, s, label='Lagrange Interpolant')\n",
    "plt.fill_between(x, s, np.zeros_like(x), alpha=0.3)\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')\n",
    "plt.title('Numerical vs. Analytical Function')\n",
    "\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  }
 ],
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