<commit_before>{
 "metadata": {
  "name": "",
  "signature": "sha256:5d1e850b91a24fefd031f93e3b8e21d9a24924d17df96d6ba85ccae88e47a311"
 },
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 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Problem Definition"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's put some heat on the MFO logo! In other words, we want to solve the stationary heat equation\n",
      "\n",
      "\\begin{align*}\n",
      "    - \\nabla \\cdot (a_{\\mu}(x)\\nabla u_\\mu(x)) &= 0 && x \\in [0, 1]^2 \\\\\n",
      "     a_{\\mu}(x) \\nabla u_\\mu(x) \\cdot n &= 1 && x \\in [0, 1] \\times \\{0\\} \\\\\n",
      "    u_\\mu(x) &= 0 && x \\in \\partial [0, 1]^2 \\setminus [0, 1] \\times \\{0\\}\n",
      "\\end{align*}\n",
      "\n",
      "for various parameters $\\mu = (\\mu_{bnd}, \\mu_{in}) \\in \\mathcal{P} := [1, 100]^2$. The diffusivity $a_\\mu$ is given by\n",
      "\n",
      "\\begin{equation}\n",
      "    a_{\\mu}(x) = a_{out}(x) + \\mu_{bnd}a_{bnd}(x) + \\mu_{in}a_{in}(x)\n",
      "\\end{equation}\n",
      "\n",
      "where $a_{out}(x) \\in [0, 100]$, $a_{bound}(x), a_{in}(x) \\in [0, 1]$ are defined by the following grayscale images with black indicating the lowest and white indicating the largest possible value:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from IPython.display import HTML\n",
      "HTML(\"\"\"<table><tr>\n",
      "<td><img src=\"mfo_outer.png\"><td><img src=\"mfo_boundary.png\"><td><img src=\"mfo_inner.png\">\n",
      "</tr></table>\"\"\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<table><tr>\n",
        "<td><img src=\"mfo_outer.png\"><td><img src=\"mfo_boundary.png\"><td><img src=\"mfo_inner.png\">\n",
        "</tr></table>"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 1,
       "text": [
        "<IPython.core.display.HTML at 0x7f33a212aad0>"
       ]
      }
     ],
     "prompt_number": 1
    },<commit_msg>fix image urls for older ipython versions<commit_after>{
 "metadata": {
  "name": "",
  "signature": "sha256:67abd7367b9b7aa1a5ada0886f19154f2c8cca9b75e22a8ee168ab2e5ca0ddb1"
 },
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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Problem Definition"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's put some heat on the MFO logo! In other words, we want to solve the stationary heat equation\n",
      "\n",
      "\\begin{align*}\n",
      "    - \\nabla \\cdot (a_{\\mu}(x)\\nabla u_\\mu(x)) &= 0 && x \\in [0, 1]^2 \\\\\n",
      "     a_{\\mu}(x) \\nabla u_\\mu(x) \\cdot n &= 1 && x \\in [0, 1] \\times \\{0\\} \\\\\n",
      "    u_\\mu(x) &= 0 && x \\in \\partial [0, 1]^2 \\setminus [0, 1] \\times \\{0\\}\n",
      "\\end{align*}\n",
      "\n",
      "for various parameters $\\mu = (\\mu_{bnd}, \\mu_{in}) \\in \\mathcal{P} := [1, 100]^2$. The diffusivity $a_\\mu$ is given by\n",
      "\n",
      "\\begin{equation}\n",
      "    a_{\\mu}(x) = a_{out}(x) + \\mu_{bnd}a_{bnd}(x) + \\mu_{in}a_{in}(x)\n",
      "\\end{equation}\n",
      "\n",
      "where $a_{out}(x) \\in [0, 100]$, $a_{bound}(x), a_{in}(x) \\in [0, 1]$ are defined by the following grayscale images with black indicating the lowest and white indicating the largest possible value:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from IPython.display import HTML\n",
      "HTML(\"\"\"<table><tr>\n",
      "<td><img src=\"files/mfo_outer.png\"><td><img src=\"files/mfo_boundary.png\"><td><img src=\"files/mfo_inner.png\">\n",
      "</tr></table>\"\"\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<table><tr>\n",
        "<td><img src=\"files/mfo_outer.png\"><td><img src=\"files/mfo_boundary.png\"><td><img src=\"files/mfo_inner.png\">\n",
        "</tr></table>"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 1,
       "text": [
        "<IPython.core.display.HTML at 0x7f2da87acad0>"
       ]
      }
     ],
     "prompt_number": 1
    },
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{'dir': 'git-commits-cleaned', 'id': '1960739', 'n_tokens_mistral': 1813, 'n_tokens_neox': 1707, 'n_words': 348}