lecture-2.ipynb

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    "# Lecture 2: Everything You Always Wanted to Know About Differential Equations*\n",
    "(*But Were Afraid to Ask)"
   ]
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    "### To be learned:\n",
    "\n",
    "* What is a PDE\n",
    "* Derive a DE (differential equation) for a system of masses and springs\n",
    "* finite difference method (FDM)\n",
    "* order of approximation\n",
    "* how to numerically solve a DE "
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   "source": [
    "## Why PDEs?\n",
    "\n",
    "PDEs (Partial Differential Equations) are imporant because:\n",
    "\n",
    "* modeling almost any physical phenomenon involves differential equations\n",
    "* almost any other physical phenomena involve integral equations\n"
   ]
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   "source": [
    "## Typical example of using numerical methods for PDEs\n",
    "\n",
    "<img width=\"700\" src=\"fig/conn_rods.png\" />\n",
    "\n",
    "[<a href=\"http://www.ima.umn.edu/~arnold//8445.f11/notes.pdf#page=8\">source</a>]"
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   "source": [
    "### Another example of using numerical methods for PDEs\n",
    "\n",
    "### where an error caused $700 million\n",
    "\n",
    "<img width=\"700\" src=\"fig/failure-2.png\" />\n",
    "\n",
    "[<a href=\"http://www.ima.umn.edu/~arnold//disasters/sleipner.html\">more details</a>] [<a href=\"http://www.ima.umn.edu/~arnold//8445.f11/notes.pdf#page=9\">source</a>]"
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   "source": [
    "### In image processing\n",
    "\n",
    "Check out an interesting lecture <a href=\"http://cvrr.ucsd.edu/eshed/class/PDEsandIP.pdf#page=6\">here</a>"
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    "## Basics\n",
    "\n",
    "### Derivative\n",
    "\n",
    "You should know what is:\n",
    "\n",
    "* a <a href=\"http://en.wikipedia.org/wiki/Derivative\">derivative</a>, and\n",
    "* a <a href=\"http://en.wikipedia.org/wiki/Partial_derivative\">partial derivative</a>."
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   "source": [
    "### PDEs\n",
    "\n",
    "* PDE = equation involving partial derivatives (see also an ODE).\n",
    "\n",
    "* An examples of a PDE:\n",
    "\n",
    "$$ \\frac{\\partial u}{\\partial t} = \\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2}, $$\n",
    "\n",
    "where $u(t,x,y)$ is an unknown function (e.g., the temperature) that depends on time $t$ and space $x,y$."
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    "### Modeling with DEs (Differential Equations)\n",
    "\n",
    "Let us model stretching a guitar string:\n",
    "<img width=\"300\" src=\"fig/guitar_string_stretch.jpg\">\n",
    "\n",
    "Consider a system of $N$ springs and $N+1$ masses:\n",
    "\n",
    "<img width=\"400\" src=\"fig/springs.png\">\n",
    "$i$-th mass is being pulled with the force $F_i$.\n",
    "\n",
    "\n",
    "* distance between masses: $h = 1/N$ (assuming that the length of the string is $1$).\n",
    "* Positions of the masses are $x_i = i h$ fixed, and $y_i$ to be found.\n",
    "* Elongation of each spring is $e_i = ((x_{i+1}-x_i)^2+(y_{i+1}-y_i)^2)^{1/2} = (h^2+(y_{i+1}-y_i)^2)^{1/2}$.\n",
    "* Assuming that all $y_i$ are small (compared to $h$), $e_i \\approx h (1+(y_{i+1}-y_i)^2/(2 h^2))$\n",
    "* Recall that the energy of a spring is\n",
    "$$\n",
    "k e_i^2/2\n",
    "\\approx \\frac{k h^2}{2} (1+(y_{i+1}-y_i)^2/(2 h^2))^2\n",
    "\\approx \\frac{k h^2}{2} (1+(y_{i+1}-y_i)^2/(h^2))\n",
    "$$\n",
    "where $k$ is the spring constant.\n",
    "* Total energy:\n",
    "$$\n",
    "E = \\frac{k h}{2} + \\frac{k h^2}{2} \\sum_{i=0}^{N-1} (y_{i+1}-y_i)^2/(h^2)\n",
    "$$\n",
    "\n",
    "* Internal force on $j$-th mass:\n",
    "$$\n",
    "f_j = -\\frac{\\partial E}{\\partial y_j} = k \\big(\n",
    "\t(y_{j+1}-y_j) - (y_{j}-y_{j-1}) \\big)\n",
    "= k  (y_{j+1}-2 y_j + y_{j-1})\n",
    "$$\n",
    "\n",
    "* Equilibrium (force balance):\n",
    "$$\n",
    "F_j = -f_j = k (-y_{j+1}+2 y_j - y_{j-1})\n",
    "$$\n",
    "\n",
    "* We know that if $y_j = u(j h)$, where $u$ is a smooth function then\n",
    "$$\n",
    "0 = h^{-2} (y_{j+1}-2 y_j + y_{j-1}) \\approx u''(hj).\n",
    "$$\n",
    "Hence if also $F_j = h^{-2} F(j h)$ then we have\n",
    "$$\n",
    "\\frac{k}{2} u'' = F\n",
    "$$\n",
    "as a model.\n",
    "\n",
    "* **Note**: The way we scaled $F$ as $h\\to 0$ is not right. To make it cleaner, we should have assumed that $k$ depends on $h$ as $c h^{-2}$."
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    "### a PDE\n",
    "\n",
    "<img width=\"400\" src=\"fig/springs_sheet.png\">\n",
    "\n",
    "* Force balance:\n",
    "$$\n",
    "h^{-2} (u_{i+1,j} - 2 u_{i,j} + u_{i-1,j} + u_{i,j+1} - 2 u_{i,j} + u_{i,j-1})\n",
    "= -F_{i,j}\n",
    "$$\n",
    "which approximates\n",
    "$$\n",
    "\\Delta u = -F,\n",
    "$$\n",
    "where $\\Delta = \\frac{\\partial^2}{\\partial x^2} + \\frac{\\partial^2}{\\partial y^2}$\n",
    "\n",
    "* This equation models a membrane.\n",
    "* This is called a **Poisson equation**\n",
    "* $\\Delta$ is called a **Laplacian**"
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    "### A boundary-value problem (BVP)\n",
    "\n",
    "To solve a PDE we need:\n",
    "* A domain $\\Omega$ (say, a unit square)\n",
    "* **Boundary conditions**. They describe what we do with the boundary masses.\n",
    "* **Dirichlet**: fixing the positions of masses (often to be zero): $u|_{\\Gamma} = 0$ ($\\Gamma = \\partial\\Omega$ is the boundary of the domain $\\Omega$)\n",
    "* **Neumann** (also known as free): leaving their (vertical) position free. This corresponds to setting $\\frac{\\partial u}{\\partial n} = 0$ on $\\Gamma$.\n",
    "\n",
    "* In general, one may have a non-zero RHS (right-hand side), e.g., $u|_{\\partial\\Omega} = g$, where $g$ is a known function."
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    "### Example of BPV\n",
    "\n",
    "$$\n",
    "\\begin{align*}\n",
    "\\Delta u = f\\qquad&\\text{on }\\Omega\n",
    "\\\\\n",
    "u = 0\\qquad&\\text{on }\\Gamma\n",
    "\\end{align*}\n",
    "$$\n",
    "(here $f$ is the new notation, nothing to do with the force from above)"
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   "source": [
    "* What does this equation describe?"
   ]
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   "source": [
    "### Solving a BVP numerically\n",
    "\n",
    "Steps:\n",
    "* Discretize a function (and a domain)\n",
    "* Discretize the equations\n",
    "* Solve"
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   "source": [
    "## Solving a BVP with Finite Differences\n",
    "\n",
    "### Discretize a domain:\n",
    "\n",
    "* Choose $h$---grid step size\n",
    "\n",
    "* Grid is points \"that are multiples\" of $h$ (also called a *mesh*).\n",
    "We denote it as $${\\mathbb R}^2_h := \\{(x_1,x_2) : x_1/h\\in{\\mathbb R}, x_2/h\\in{\\mathbb R}\\}$$\n",
    "\n",
    "* Discrete domain:\n",
    "$$\n",
    "\\Omega_h := \\Omega \\cap {\\mathbb R}^2_h\n",
    "$$\n",
    "\n",
    "* Discrete boundary:\n",
    "$$\n",
    "\\Gamma_h := \\Gamma \\cap {\\mathbb R}^2_h\n",
    "$$\n",
    "\n",
    "<center><img width=200 src=\"fig/grid.png\"></center>\n",
    "\n",
    "<center>Black points = $\\Omega_h$, <span style=\"color: blue\">blue points</span> = $\\Gamma_h$</center>"
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   "source": [
    "* Why \"multiples\" are in quotes?"
   ]
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   "source": [
    "* How to read the definition of $\\Omega_h$?"
   ]
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   "source": [
    "### Discretize a function:\n",
    "\n",
    "* A continuous function $u(x,u)$ is approximated by $u_h$ with values on $(x,y)\\in\\Omega_h$."
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    "### Discretize the equation:\n",
    "\n",
    "* We need to discretize the operator $\\Delta = \\frac{\\partial^2}{\\partial x^2} + \\frac{\\partial^2}{\\partial y^2}$\n",
    "* Recall that for $g=g(x)$,\n",
    "$$\n",
    "\\frac{d^2 f}{dx^2} = \\frac{g(x-h)-2g(x)+g(x+h)}{h^2} + O(h^2)\n",
    "$$\n",
    "\n",
    "* Exercise: prove it by Taylor series:\n",
    "$$\n",
    "g(x+\\xi) = g(x) + \\xi g'(x) + \\frac{\\xi^2}{2} g''(x) + \\frac{\\xi^3}{6} g'''(x) + O(\\xi^4)\n",
    "$$\n",
    "<small>(refresh the <a href=\"http://en.wikipedia.org/wiki/Big_O_notation\">Big-O</a> notation if you need)</small>\n",
    "\n",
    "* Hence\n",
    "$$\n",
    "\\Delta u = \\Delta_h u + O(h^2),\\qquad\\text{where}\n",
    "$$\n",
    "$$\n",
    "\\Delta_h u := \\frac{u(x-h,y)+u(x+h,y)+u(x,y+h)+u(x,y-h)-4u(x,y)}{h^2}.\n",
    "$$\n",
    "\n",
    "* Thus, the **discretized equation** is\n",
    "$$\n",
    "\\frac{u(x-h,y)+u(x+h,y)+u(x,y+h)+u(x,y-h)-4u(x,y)}{h^2} = f(x,y)\n",
    "\\qquad (x,y)\\in\\Omega_h\n",
    "$$\n",
    "\n",
    "* Discrete boundary conditions are easy:\n",
    "$$\n",
    "u(x,y) = 0\n",
    "\\qquad (x,y)\\in\\Gamma_h\n",
    "$$\n",
    "\n",
    "* This is a **system of linear equations SLE** on $u(x,y)$ for $(x,y)\\in\\Omega_h\\cap\\Gamma_h$.\n",
    "\n",
    "* Exercise: Let $h=1/n$ how many unknowns and equations are there?"
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    "### Order of Approximation\n",
    "\n",
    "* Because the error is $O(h^2)$, we say that $\\Delta_h u$ is approximates $\\Delta u$ with the **second order**.\n",
    "* In othre words, we have the **second order of approximation**."
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    "### SLE\n",
    "\n",
    "* Let us do it for $n=4$. We have 9 unknowns (Exercise: write them down)\n",
    "* It is easier to denote them $u_{1,1},u_{1,2},\\ldots,u_{3,3}$ (where $u_{i,j} = u(h i, h j)$).\n",
    "* We have 9 equations of the form $\\Delta_h u(x,y)=0$, $(x,y)\\in\\Omega_h$.\n",
    "* If we collect the unknowns in a vector, we will have a linear system\n",
    "\\\\[\n",
    "\\frac{1}{h^2}\n",
    "\\begin{pmatrix}\n",
    "-4 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    " 1 &-4 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n",
    " 0 & 1 &-4 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n",
    " 1 & 0 & 0 &-4 & 1 & 0 & 1 & 0 & 0 \\\\\n",
    " 0 & 1 & 0 & 1 &-4 & 1 & 0 & 1 & 0 \\\\\n",
    " 0 & 1 & 0 & 0 & 1 &-4 & 0 & 1 & 0 \\\\\n",
    " 0 & 0 & 0 & 1 & 0 & 0 &-4 & 1 & 0 \\\\\n",
    " 0 & 0 & 0 & 0 & 1 & 0 & 1 &-4 & 1 \\\\\n",
    " 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 &-4 \\\\\n",
    "\\end{pmatrix}\n",
    "\\begin{pmatrix}\n",
    "u_{1,1} \\\\ u_{1,2} \\\\ u_{1,3} \\\\ u_{2,1} \\\\ u_{2,2} \\\\ u_{2,3} \\\\ u_{3,1} \\\\ u_{3,2} \\\\ u_{3,3}\n",
    "\\end{pmatrix}\n",
    "=\n",
    "\\begin{pmatrix}\n",
    "f_{1,1} \\\\ f_{1,2} \\\\ f_{1,3} \\\\ f_{2,1} \\\\ f_{2,2} \\\\ f_{2,3} \\\\ f_{3,1} \\\\ f_{3,2} \\\\ f_{3,3}\n",
    "\\end{pmatrix}\n",
    "\\\\]\n",
    "\n",
    "* Exercise: guess how $f_{i,j}$ is defined"
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    "#### Block structure\n",
    "\n",
    "* The matrix has a block form\n",
    "$$\n",
    "\\begin{pmatrix}\n",
    "A & I & O \\\\\n",
    "I & A & I \\\\\n",
    "O & I & A\n",
    "\\end{pmatrix}\n",
    ",\n",
    "$$\n",
    "where $I$ is the 3x3 identity matrix, $O$ is the zero matrix, and\n",
    "$$A = \\begin{pmatrix}\n",
    "-4 & 1 & 0 \\\\\n",
    " 1 &-4 & 1 \\\\\n",
    " 0 & 1 &-4\n",
    "\\end{pmatrix}\n",
    "$$\n",
    "\n",
    "* In general, the matrix contains $(n-1)\\times(n-1)$ blocks (most of the blocks are zero), each block is an $(n-1)\\times(n-1)$ matrix:\n",
    "$$\n",
    "\\frac{1}{h^2}\n",
    "\\begin{pmatrix}\n",
    "A & I & O & \\cdots & O & O\\\\\n",
    "I & A & I & \\cdots & O & O\\\\\n",
    "O & I & A & \\cdots & O & O\\\\\n",
    "\\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n",
    "O & O & O & \\cdots & A & I \\\\\n",
    "O & O & O & \\cdots & I & A \\\\\n",
    "\\end{pmatrix}\n",
    ",\n",
    "$$\n",
    "And the $(n-1)\\times (n-1)$ matrix $A$ has a tridiagonal structure with $-4$ on the diagonal and $1$ above and below the diagonal.\n"
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   "source": [
    "** TODO **: write a python code that solves and plots the solution for $f\\equiv 1$. Would be good to use the Kronecker product"
   ]
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x16b3e518>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
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       "LumJSHceFHpJl3RJl3SKdIHVJV3SJT0R6QKrS7qkS3oi0gVWl3RJl/REpAusLumSLumJSBdYXdIl\n",
       "XdITkS6wuqRLuqQnIl1gdUmXdElPRPr/sH+Hx40wXUIAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x16b49470>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import scipy as sp\n",
    "import matplotlib.cm as cm\n",
    "%matplotlib inline\n",
    "\n",
    "#Problem size:\n",
    "N = 15\n",
    "\n",
    "# Making A without Kronecker product:\n",
    "def lap2d(N):\n",
    "    import  pyamg\n",
    "    return sp.sparse.csr_matrix(pyamg.gallery.laplacian.poisson((N,N)))\n",
    "A = lap2d(N)\n",
    "\n",
    "# Making A with kronecker product\n",
    "# Making 1D operator:\n",
    "B = np.diag(2*np.ones(N))  + np.diag((-1)*np.ones(N-1),k=-1)+ np.diag((-1)*np.ones(N-1),k = 1)\n",
    "Id = np.diag(np.ones(N));\n",
    "# Making 2D operator:\n",
    "A = np.kron(Id,B) + np.kron(B,Id)\n",
    "\n",
    "# Fig. of A\n",
    "plt.spy(A,markersize=20/N)\n",
    "\n",
    "# Right hand side\n",
    "f = np.ones(N**2)\n",
    "\n",
    "# Solution\n",
    "x = sp.sparse.linalg.spsolve(A,f)\n",
    "Sq_x = x.reshape((N, N))\n",
    "#print Sq_x\n",
    "\n",
    "#Fig. of solution\n",
    "fig, ax = plt.subplots()\n",
    "\n",
    "im = ax.imshow(Sq_x, cmap=cm.RdBu, vmin=abs(Sq_x).min(), vmax=abs(Sq_x).max(), extent=[0, 1, 0, 1])\n",
    "im.set_interpolation('bilinear')\n",
    "\n",
    "cb = fig.colorbar(im, ax=ax)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Methods of solving the SLE\n",
    "\n",
    "* Full Gaussian Elimination: $O(N^3)$---no one does that anymore ($N=n^2$ in 2D and $N=n^3$ in 3D)\n",
    "* Elimination using sparsity: $O(N^{3/2})$ in 2D---ok; $O(N^2)$ in 3D---too much!\n",
    "* Fast Fourier Transform: $O(N \\log(N))$---very good, but not always applicable (requires a square domain)\n",
    "* Multigrid: $O(N)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Suggested literature:\n",
    "\n",
    "* <a href=\"http://www.ima.umn.edu/~arnold//8445.f11/notes.pdf\">Lecture notes on Numerical Analysis of Partial Differential Equations</a>, Chapters 1 and 2. (Beware: mathematics.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "##### Questions?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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    "collapsed": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "from IPython.core.display import HTML\n",
    "def css_styling():\n",
    "    styles = open(\"./styles/custom.css\", \"r\").read()\n",
    "    return HTML(styles)\n",
    "css_styling()\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [],
   "source": []
  }
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