lecture-1.ipynb

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    "# Lecture 1: Differential Equations"
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   "source": [
    "## Why computer modelling? \n",
    "Because it is **cheaper** than real-life experiment, or in the case when real-life experiment **is not possible**."
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    "## Typical steps of computer modelling\n",
    "\n",
    "1. Write down **equations**\n",
    "\n",
    "2. **Discretize**: Replace the continious problem by a discrete problem\n",
    "    * Discretize unknowns\n",
    "    * Discretize equations\n",
    "\n",
    "3. **Solve** the resulting discrete problem\n"
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    "## Equations:\n",
    "\n",
    "What kind of equations are used?\n",
    "\n",
    "* Differential equations\n",
    "\n",
    "* ?\n",
    "\n",
    "* ?\n"
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    "## 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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    "### 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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    "### In image processing\n",
    "\n",
    "Check out an interesting lecture <a href=\"https://www.lri.fr/~gcharpia/VisionSeminar/slides/2014-04-02-kornprobst-vist.pdf\">here</a>"
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   "source": [
    "## Discretization\n",
    "The discretization is the replacement of the region by discrete elements:\n",
    "<img src=\"pic/airplane2.jpeg\" style=\"width: 700px;\" >"
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    "## Random notes\n",
    "- Discretization and solve can be connected\n",
    "- We'll use standard solvers. Ivan Oseledets will read a course on efficient solvers\n",
    "- Only a subproblem in **design and optimization**\n",
    "- Many physical problems are still too complex for math (turbulence, quantum mechanics)"
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    "Many **process in physics** are modelled as PDEs.\n",
    "\n",
    "- Diffusion processes (heat transfer), electrostatics (circuit design) Poisson equation \n",
    "- Sound propagation (noise on the streets, buildings) -- Helmholtz\n",
    "equation \n",
    "- Electromagnetics -- fMRI (functional magnetic resonance imaging) -\n",
    "Maxwell equations \n",
    "- Fluid flows -- Euler / Navier Stokes/ Stokes / equations "
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    "Some are modelled with ODEs:\n",
    "\n",
    "- Molecular dynamics -- Newton equations\n",
    "\n",
    "Some are algebraic equations:\n",
    "\n",
    "- Molecular statics: protein confirmations -- potential energy minimization"
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    "PDEs (ODEs) appear in:\n",
    "- Financial math (Black Scholes equation) \n",
    "- Chemical engineering (Smoluchowsky equation)\n",
    "- Nanoworld (Schrodinger equation) \n",
    "- Biology (populations dynamics equations)"
   ]
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    "## What we plan to cover:\n",
    "- **Equations** and **discretization**\n",
    "\n",
    "- a bit of modeling\n",
    "\n",
    "- Start thinking about the App Period projects **now**\n",
    "    - you **may** do one App Period project for both courses you take, subject to my approval"
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    "## Software packages\n",
    "\n",
    "Unless you need something very special, you'd not implement numerical methods from scratch. There are free and commercial packages for you.\n",
    "\n",
    "* Most are based on FEM\n",
    "* Fluid (=liquid and gases) mechanics is often FVM\n",
    "\n",
    "An example of commercial package is [Abaqus](https://en.wikipedia.org/wiki/Abaqus)"
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    "**Free FEM software packages**\n",
    "(probably outdated)\n",
    "\n",
    "- [FEniCS project](http://fenicsproject.org/) - beautiful finite element package, superhard to install \n",
    "- [FiPy](http://www.ctcms.nist.gov/fipy/) - Finite volume solver\n",
    "- [deal.II](https://www.dealii.org/) - FEM solver, C++ (with some Python interface)\n",
    "- [DUNE](http://dune.mathematik.uni-freiburg.de/) - FEM solver"
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    "**Word of wisdom**\n",
    "\n",
    "- Packages sometimes do not give you the right answer (e.g., smear out small details just to be safe)\n",
    "- You should understand what you are doing (as much as you can)\n",
    "- Some principles of numerical analysis are universal -- they may help even you do not completely understand what you are doing."
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    "**Mathematica demo**\n",
    "\n",
    "[demo](Math-DE-demo.nb)\n",
    "\n",
    "[further materials](https://www.wolfram.com/mathematica/new-in-10/pdes-and-finite-elements/)"
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    "**A commercial package (ANSYS) ad**\n",
    "\n",
    "[Digital twin](https://www.youtube.com/watch?v=cxoHx7kitzk)"
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    "## Notes on Modeling\n",
    "**(aka what is similar between PDEs and machine learning)**\n",
    "\n",
    "* A PDE model: $u _ t = u _ {xx} + u _ {yy} + f(x)$\n",
    "* A numerical model:\n",
    "<img src=\"pic/heat_sol.jpg\" style=\"width: 200px;\"/>\n",
    "* A data-driven (machine-learning) model: <img src=\"pic/NN.jpg\" style=\"width: 200px;\"/>"
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   "source": [
    "# For the next lecture:\n",
    "\n",
    "* Do the following 2 exercises:"
   ]
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    "## Problem of interpolation\n",
    "\n",
    "* Given a function $f(x)$ to be interpolated on an interval $[a,b]$ and interpolation points $x _i$, $a \\leq x _0 < x _1 < ... < x _n \\leq b$, find $\\phi(x) \\approx f(x)$. You can only use values of $f$ at $x _i$.\n",
    "\n",
    "* Exercise 1: $\\phi(x) = const$, $x_0=a$. Express the error $f(x) - \\phi(x)$ as $f'(\\xi) (x-x_0)$, where $\\xi$ lies between $x_0$ and $x$. **Hint:** use the mean-value theorem or recall Taylor series.\n",
    "\n",
    "* Exercise 2: $\\phi(x) = c_0 + c_1 x$, $x_0=a$, $x_1 = b$. Find a good expression for the error $f(x) - \\phi(x)$."
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    "also we'll cover\n",
    "\n",
    "## Quadrature\n",
    "\n",
    "* Given a function $f(x)$, to be interpolated on an interval $[a,b]$ and interpolation points $x _i$, $a \\leq x _0 < x _1 < ... < x _n \\leq b$, find $c_0, ..., c_n$ such that\n",
    "$$ \\int _a ^b f(x) dx \\approx \\sum _i c_i f(x_i) $$\n",
    "\n",
    "[//]: # \"* Exercise 3: For $n=1$ (two-point integration), $a=-h$, $b=h$, express the error as $O(h^k)$ and find $x_0$ and $x_1$ that optimize the error, i.e., maximize $h$.\""
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