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            <!--span><img src="images/google_developers_logo.png"></span-->
            <span><img src="images/matheon-w-text.svg" alt="matheon logo"></span>
        </article>
    </slide>

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        <aside class="gdbar"><img src="images/matheon-logo-color.svg" alt="mathon logo"></aside>
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            <p>
                <h2 data-config-presenter><!-- populated from slide_config.json --></h2>
        </hgroup>
    </slide>

    <slide>
        <hgroup>
            <h2>Overview</h2>
        </hgroup>
        <article>
    <img src="images/python-powered-h.svg" alt="Description" title="Description" style="float:right;margin:0 5px 0 0;" width="200px">
          <ul>
            <li>What is Python?</li>
            <ul>
            <li>Some general facts about Python</li>
            <li>Programming 101</li>
            <li>Selected packages for scientific computation:<br/>
              NumPy, SciPy, matplotlib, SymPy, FEniCS/DOLFIN
            </li>
            </ul>
          </ul>
        <footer class="source">www.python.org/community/logos/</footer>
        </article>
    </slide>

    <slide>
        <hgroup>
            <h2>What is Python?</h2>
        </hgroup>
        <article>
          <ul>
            <li>free software</li>
            <li>universal programming language</li>
            <li>many packages existing</li>
            <li>easily extendable</li>
          </ul>
          <p>Python is (line many script languages) very intuitive.</p>
          <ul>
            <li>no explicit type specifications (<a href="https://de.wikipedia.org/wiki/Duck-Typing">duck typing</a>)</li>
            <li>code blocks must be indented (leads to improved readability)</li>
            <li>Very high-quality code.<br/>
      0.005 defects per thousand lines of code:<br/>
      "The open-source Python programming language [...] now surpasses that of
its open-source and proprietary peers, according to a study published by
development testing vendor Coverity." -- Sean M. Kerner, eweek.com
</li>
          </ul>
        </article>
    </slide>

    <slide>
      <hgroup>
        <h2>The Python language: Popularity</h2>
      </hgroup>
      <article class="flexbox vcenter">
        <img src="images/tpci_trends.png" alt="Description" title="Description" width=700px>
        <footer class="source">source: http://www.tiobe.com/</footer>
      </article>
    </slide>

    <slide>
      <hgroup>
        <h2>10 Reasons Python Rocks for Research</h2>
      </hgroup>
      <article>
      <ul>
      <li>Holistic Language Design<br/>
      "Doing mathematical programming in a general-purpose language is
easier than doing general-purpose programming in a mathematical language."
      </li>
      <li>Readability</li>
      <li>Balance of High Level and Low Level Programming</li>
      <li>Language Interoperability</li>
      <li>Documentation System</li>
      <li>Hierarchical Module System</li>
      <li>Data Structures</li>
      <li>Available Libraries</li>
      <li>Testing Framework</li>
      <li>Large user base</li>
      </ul>
      Possible downsides?
      <ul>
      <li>Not as popular as, e.g., MATLAB(r)<br/>
      <li>...<br/>
      </ul>
      </article>
        <footer class="source">http://www.stat.washington.edu/~hoytak/blog/whypython.html</footer>
    </slide>

    <slide>
      <hgroup>
        <h2>The language: "Hello world."</h2>
      </hgroup>
      <article>
        <pre class="prettyprint" data-lang="python">
print('Hello world!')
a = 42                           # integer
b = 42 / 21 + 10                 # = 12
my_list = [1, 3, 'Hello World!'] # list: all types permitted!
print(my_list[0])                # 0-based indexing
for i in range(4):               # i in 0,...,3
    print(i)
for elem in my_list:
    print(elem)
d = { 'Hallo': 'hello',       # dictionary ("map")
      'Welt': 'world',
      42: 'fourty-two'}
print(d['Hallo'])             # -> hello
tuple = (4,3,1)
if tuple == (4,):             # easy to compare!
    print('yeah')
</pre>
      </article>
    </slide>

    <slide>
      <hgroup>
        <h2>The language: functions, modules</h2>
      </hgroup>
      <article>
        <pre class="prettyprint" data-lang="python">
# mymodule.py
def square(a):       # functions can be defined anywhere
    return a*a       # with arbitrary names (like variables)

def arnoldi(A, v, maxiter=10, ortho='mgs'):  # default arguments
    # compute V, H
    return V, H      # multiple return arguments
</pre>
        <pre class="prettyprint" data-lang="python">
import mymodule      # import is possible anywhere in the code
print( mymodule.square(3) )
</pre>
        <pre class="prettyprint" data-lang="python">
from mymodule import square, arnoldi
print( square(3) )
V, H = arnoldi(A, v, ortho='house') # maxiter is set to default (10)
</pre>
      </article>
    </slide>

  <slide class="fill nobackground" style="background-image: url(images/package.jpg)">
    <h2 class="white">Packages (modules, add-ons,...) </h2>
    <footer class="source">source: flickr.com/photos/halfbisqued/</footer>
  </slide>

    <slide>
      <hgroup>
        <h2>Packages: NumPy</h2>
      </hgroup>
      <article>
        <p>NumPy is <em>the</em> basis package for numerics (based on BLAS,
        LAPACK, etc.).</p>
        <pre class="prettyprint" data-lang="python">
from numpy import array, ones, dot   # N-dim Arrays
v = array([1,2,3])           # 1-dim arr, shape==(3,)
v = ones(5)                  # 1-dim arr, shape==(5,)
v = ones((5,1))              # 2-dim arr, shape==(5,1)
A = array([[1,2,3],[4,5,6]]) # 2x3-matrix: 2-dim arr, shape==(2,3)
A = ones((5,5))              # 5x5-matrix: 2-dim arr, shape==(5,5)
b = dot(A, v)                # A*v
</pre>
        <p>The default data type in NumPy are <em>NumPy arrays</em>; there
        are also <em>NumPy matrices</em>. The main difference is that one can
        use the *-operator for matrix-matrix multiplication (as in MATLAB).
        Many (especially the <a href="http://wiki.scipy.org/NumPy_for_Matlab_Users">NumPy community</a>) recommend <em>NumPy arrays</em>.</p>
      </article>
      <aside class="gdbar right bottom"><img src="images/numpylogo_med.png"></aside>
    </slide>

    <slide>
      <hgroup>
        <h2>Packages: NumPy (cont'd)</h2>
      </hgroup>
      <article>
        <p>Well-known commands in NumPy:</p>
        <ul>
          <li>zeros, ones, diag, abs, linspace, ...</li>
          <li><strong>module random:</strong> rand, randn, ...</li>
          <li><strong>module linalg:</strong> solve, norm, eig, svd, lu, qr, cholesky, ...</li>
          <li><strong>module polynomial:</strong> monomial, Chebyshev, Legendre, ...</li>
        </ul>

        <p>MATLAB's \-operator: solve(A,b).</p>
        <p>For more info, see <a href="http://docs.scipy.org/doc/NumPy/reference/">NumPy documentation</a>.</p>
      </article>
      <aside class="gdbar right bottom"><img src="images/numpylogo_med.png"></aside>
    </slide>

    <slide>
      <hgroup>
        <h2>Packages: SciPy</h2>
      </hgroup>
      <article>
        <iframe src="http://docs.scipy.org/doc/scipy/reference/"></iframe>
      </article>
    </slide>

    <slide>
      <hgroup>
        <h2>Packages: SciPy (cont'd)</h2>
      </hgroup>
      <article>
        <p>Interesting for us:</p>
        <ul>
          <li><strong>scipy.linalg:</strong> extended functionality: banded matrices, qz, matrix functions, ...</li>
          <li><strong>scipy.sparse:</strong> several sparse matrix types, cg, gmres, eigs, ...</li>
          <li><strong>scipy.optimize:</strong> simplex, newton, leastsq, ...</li>
        </ul>

        <p>For more info, see <a href="http://docs.scipy.org/doc/scipy/reference/">SciPy documentation</a>.</p>
      </article>
      <aside class="gdbar right bottom"><img src="images/scipyshiny_small.png"></aside>
    </slide>

    <slide>
      <hgroup>
        <h2>Packages: matplotlib</h2>
      </hgroup>
      <article>
        <iframe src="http://matplotlib.org/gallery.html" ></iframe>
      </article>
    </slide>

    <slide>
      <hgroup>
        <h2>Packages: matplotlib (cont'd)</h2>
      </hgroup>
      <article>
        <pre class="prettyprint" data-lang="python">
from matplotlib import pyplot as pp

data = [x**2 for x in range(100)] # list comprehension!
pp.plot(data)
pp.show()
</pre>
      <p>LaTeX-integration via <a href="https://github.com/nschloe/matplotlib2tikz">matplotlib2tikz</a>.</p>
      </article>
      <aside class="gdbar right bottom"><img src="images/matplotlib_logo.png"></aside>
    </slide>

    <slide>
      <hgroup>
        <h2>Packages: SymPy</h2>
      </hgroup>
      <article>
        Symbolic calculations in Python.<br/>
        <pre class="prettyprint" data-lang="python">
import sympy as smp

x, y = smp.symbols('x, y')

f = x**2 + sin(x) * y
diff(f, x)

diff(f, x, 2)
</pre>
     full-featured computer algebra system<br/>
     <ul>
     <li>differentiation</li>
     <li>integration</li>
     <li>...</li>
     </ul>
      </article>
    <aside class="gdbar right bottom"><img src="images/sympylogo.png"></aside>
    </slide>

    <slide>
      <hgroup>
        <h2>Packages: IPython</h2>
      </hgroup>
      <article>
      IPython provides a rich architecture for interactive computing with:<br/>
    <img src="images/ipy_0.13.png" style="float:right;margin:0 5px 0 0;" width="400px">
      <ul>
<li>Powerful interactive shells (terminal and Qt-based).</li>
<li>A browser-based notebook with support for code, text, mathematical expressions, inline plots and other rich media.</li>
<li>Support for interactive data visualization and use of GUI toolkits.</li>
<li>Flexible, embeddable interpreters to load into your own projects.</li>
<li>Easy to use, high-performance tools for parallel computing.</li>
      </ul>
      </article>
      <aside class="gdbar right bottom"><img src="images/ipython_logo.png"></aside>
    <footer class="source">source: ipython.org</footer>
    </slide>

    <slide>
      <hgroup>
        <h2>Packages: Miscellaneous</h2>
      </hgroup>
      <article>
        <ul>
          <li><a href="https://github.com/andrenarchy/krypy">krypy</a> (by André Gaul, TU Berlin) - Krylov subspace methods &amp; tools</li>
    <img src="images/minres.png" width="300px">
          <li><a href="https://code.google.com/p/pyamg/">pyamg</a> - algebraic multigrid</li>
    <img src="images/rootnode_aggregation.png"  width="200px">
          <li>...</li>
        </ul>
      </article>
    </slide>

  <!--slide class="fill nobackground" style="background-image: url(images/fenics.png)"-->
  <slide>
    <article class="flexbox vcenter">
    <h2>Numerical PDE solving with</h2><br/><br/>
    <img src="images/fenics.png" width="600px">
    </article>
    <footer class="source">source: fenics-project.org</footer>
  </slide>

  <slide>
    <hgroup>
      <h2>Solving PDEs with FEM</h2>
    </hgroup>
    <article>
    \[
    -\Delta u = f
    \]
    \[
    -\int_{\Omega} (\Delta u) v = \int_{\Omega} f v
    \]
    \[
    \int_{\Omega} \nabla u \cdot \nabla v - \int_{\partial\Omega} (\n\cdot\nabla u) v= \int_{\Omega} f v
    \]
    With \(v\in V^{(h)}\) (test functions), \(u = \sum_j \alpha_j u_j\), \(u_j\in U^{(h)}\) (trial functions), one gets
    an equation system for \(\alpha_i\):
    \[
    \sum_j \alpha_j \int_{\Omega} \nabla u_j \cdot \nabla v_i - \int_{\partial\Omega} (\n\cdot\nabla u_j) v_i= \int_{\Omega} f v_i \quad \forall v_i\in V^{(h)}.
    \]
    </article>
  </slide>

  <slide>
    <hgroup>
      <h2>Solving PDEs with FEM</h2>
    </hgroup>
    <article class="flexbox vcenter">
    \[
    \sum_j \alpha_j \int_{\Omega} \nabla u_j \cdot \nabla v_i - \int_{\partial\Omega} (\n\cdot\nabla u_j) v_i= \int_{\Omega} f v_i \quad \forall v_i\in V^{(h)}.
    \]
      <img src="images/ddg_hat_function.svg" alt="Description" title="Description">
      <footer class="source">source: http://brickisland.net/cs177/wp-content/uploads/2011/11/ddg_hat_function.svg</footer>
    </article>
  </slide>


  <!--slide class="segue dark quote nobackground"-->
  <slide class="fill nobackground" style="background-image: url(images/bjarne.jpg)">
    <article class="flexbox vleft auto-fadein">
      <q class="white">
        Always stay as high-level as possible.
      </q>
      <div class="author white">
        Bjarne Stroustrup<br>
        Designer of C++
      </div>
    </article>
  </slide>

  <slide>
    <hgroup>
      <h2>Common misconceptions (234): Do it yourself</h2>
    </hgroup>
    <article class="flexbox vcenter">
      <img src="images/batmobile.jpg" height="500px" alt="Description" title="Description">
      <footer class="source">source: wikia.nocookie.net</footer>
    </article>
  </slide>

  <slide>
    <hgroup>
      <h2>Writing your own FEM application</h2>
    </hgroup>
    <article>
    Compute local stiffness matrix:
    <p><pre class="prettyprint" data-lang="matlab">
J = [ X(NODES(2),1)-X(NODES(1),1) , X(NODES(3),1)-X(NODES(1),1)  ;
      X(NODES(2),2)-X(NODES(1),2) , X(NODES(3),2)-X(NODES(1),2)  ];
AbsDetJ = abs( J(1,1)*J(2,2) - J(1,2)*J(2,1) ) ;
L = (eps/(2*AbsDetJ)) * [ J(1,2)^2 + J(2,2)^2  , - J(1,1)*J(1,2) - J(2,1)*J(2,2)  ;
                                     0         ,   J(1,1)^2 + J(2,1)^2       ];
A = z3A(1,1) =   L(1,1) + 2*L(1,2) + L(2,2) + AbsDetJ/12 ;
A(1,2) = - L(1,1) - L(1,2)            + AbsDetJ/24 ;
A(1,3) = - L(1,2) - L(2,2)            + AbsDetJ/24 ;

A(2,1) = - L(1,1) - L(1,2)            + AbsDetJ/24 ;
A(2,2) =   L(1,1)                     + AbsDetJ/12 ;
A(2,3) =   L(1,2)                     + AbsDetJ/24 ;

A(3,1) = - L(1,2) - L(2,2)            + AbsDetJ/24 ;
A(3,2) =   L(1,2)                     + AbsDetJ/12 ;
A(3,3) =   L(2,2)                     + AbsDetJ/12 ;
    </pre></p>
    </article>
  </slide>

  <slide>
    <hgroup>
      <h2>Writing your own FEM application (cont'd)</h2>
    </hgroup>
    <article>
    Insert into global matrix according to Element Connectivity Table:
    <p><pre class="prettyprint" data-lang="matlab">
for r=1:size(ECT,1)
  localStiffnessMatrix = local_assemble(ECT(r,:),X,eps);
  for alpha=1:3
     i = ECT(r,alpha);
     for beta= 1:3
        j = ECT(r,beta);
        K(i,j) = K(i,j) + localStiffnessMatrix(alpha,beta);
     end
  end
end
    </pre></p>
    Still missing:
    <ul>
    <li>Quadrature of RHS</li>
    <li>boundary conditions</li>
    <li>...</li>
    </ul>
    </article>
  </slide>

  <slide>
    <hgroup>
      <h2>Do it yourself: Reality</h2>
    </hgroup>
    <article class="flexbox vcenter">
      <img src="images/makeshiftcar.jpg" height="500px" alt="Description" title="Description">
      <footer class="source">source: rofl.to</footer>
    </article>
  </slide>

  <!--slide>
    <hgroup>
      <h2>The FEniCS project</h2>
    </hgroup>
    <article>
    <ul>
    <li>Founded 2003 (Univ. of Chicago; Chalmers)</li>
    <li>People:</li>
    <ul>
    <li>Anders Logg</li>
    <li>Johan Hake</li>
    <li>Garth Wells</li>
    <li>Johannes Ring</li>
    <li>Marie Rognes</li>
    <li>...</li>
    </ul>
    <li>Involved institutions:</li>
    <ul>
    <li>Argonne National Laboratory</li>
    <li>Chalmers University of Technology</li>
    <li>Delft University of Technology</li>
    <li>Royal Institute of Technology</li>
    <li>Simula Research Laboratory</li>
    <li>University of Cambridge</li>
    <li>University of Chicago</li>
    <li>...</li>
    </ul>
    </ul>
    </article>
  </slide-->

  <!--slide>
    <hgroup>
      <h2>The FEniCS project</h2>
    </hgroup>
    <article class="flexbox vcenter">
      <img src="images/fenics-map.png" alt="Description" title="Description">
      <footer class="source">source: http://fenicsproject.org/</footer>
    </article>
  </slide-->

  <slide>
    <hgroup>
      <h2>FEniCS</h2>
    </hgroup>
    <article>
    Required: Weak formulation
    \[
    \sum_j \alpha_j \left(\int_{\Omega} \nabla u_j \cdot \nabla v_i - \int_{\partial\Omega} (\n\cdot\nabla u_j) v_i\right)= \int_{\Omega} f v_i \quad \forall v_i\in V^{(h)}.
    \]
    Then
    <p><pre class="prettyprint" data-lang="python">
from dolfin import *

# Create mesh and define function space
mesh = UnitSquareMesh(20, 20)
V = FunctionSpace(mesh, 'Lagrange', 1)

# Define boundary conditions
u0 = Expression('1 + x[0]*x[0] + 2*x[1]*x[1]')
def u0_boundary(x, on_boundary):
    return on_boundary
bc = DirichletBC(V, u0, u0_boundary)
    </pre></p>
    </article>
  </slide>

  <slide>
    <hgroup>
      <h2>FEniCS (cont'd)</h2>
    </hgroup>
    <article>
    <p><pre class="prettyprint" data-lang="python">
[...]
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
f = Constant(-6.0)
a = inner(nabla_grad(u), nabla_grad(v))*dx
L = f*v*dx

# Compute solution
u = Function(V)
solve(a == L, u, bc)

# Plot solution and mesh
plot(u)
plot(mesh)

# Dump solution to file in VTK format
file = File('poisson.pvd')
file << u

# Hold plot
interactive()
    </pre></p>
    </article>
  </slide>

    <slide>
      <hgroup>
        <h2>Result</h2>
      </hgroup>
      <article class="flexbox vcenter">
        <img src="images/fenics-output.png" width=800px>
      </article>
    </slide>

  <slide>
    <hgroup>
      <h2>Controlling the solution process</h2>
    </hgroup>
    <article>
    "Make the easy things easy and hard things possible."<br/>
    <p><pre class="prettyprint" data-lang="python">
prm = parameters['krylov_solver'] # short form
prm['absolute_tolerance'] = 0.0
prm['relative_tolerance'] = 1E-13
prm['maximum_iterations'] = 10000
prm['monitor_convergence'] = True
solve(a == L, u, [bc0, bc1],
      solver_parameters={'linear_solver': 'cg',
                         'preconditioner': 'ilu'}
      )
    </pre></p>
    </article>
  </slide>

  <slide>
    <hgroup>
      <h2>Accessing matrix, rhs</h2>
    </hgroup>
    <article>
    Choose the uBLAS backend:
    <p><pre class="prettyprint" data-lang="python">
    from dolfin import *
    parameters['linear_algebra_backend'] = 'uBLAS'
    [...]
    a = inner(nabla_grad(u), nabla_grad(v))*dx
    L = f*v*dx + g*v*ds

    A = assemble(a, bcs=[bc0,bc1])
    b = assemble(L, bcs=[bc0,bc1])
    </pre></p>
    Output:
    <p><pre class="prettyprint" data-lang="bash">
$ python ex5.py
(array([    0,     3,     8, ..., 11433, 11438, 11441]), array([   0,    1,    2, ..., 1678, 1679, 1680]), array([ 1. , -0.5, -0.5, ...,  0. ,  0. ,  1. ]))
[ 0.00114583  0.0065625   0.         ...,  0.          1.          1.        ]
    </pre></p>
    </article>
  </slide>

  <slide>
    <hgroup>
      <h2>Using a mesh</h2>
    </hgroup>
    <article>
    Typical PDE workflow:
    <img src="images/spiral.png" alt="Description" title="Description" style="float:right;margin:0 5px 0 0;" width="200px">
    <ul>
    <li>Build your mesh: Gmsh, tetgen, CUBIT, ... (formats: VTK, Exodus,...)</li>
    <li>Feed the mesh into the solver.</li>
    </ul>
    <br/>
    <br/>
    For your <b>convenience</b>, FEniCS contains simple mesh generators.
    <p><pre class="prettyprint" data-lang="python">
UnitSquare(4, 4)
UnitCube(4, 4, 4)
Rectangle(a, 0, b, 1, nr, nt, 'crossed')
[...]
    </pre></p>
     <footer class="source">image source: http://geuz.org/gmsh/</footer>
    </article>
  </slide>

  <slide>
    <hgroup>
      <h2>Using a mesh (cont'd)</h2>
    </hgroup>
    <article>
    Convert any mesh into Dolfin's format:
    <p><pre class="prettyprint" data-lang="bash">
    $ dolfin-convert mymesh.msh mymesh.xml
    dolfin-convert mymesh.msh mymesh.xml
    Converting from Gmsh format (.msh, .gmsh) to DOLFIN XML format
    Expecting 2001 vertices
    Found all vertices
    Expecting 10469 cells
    Found all cells
    Conversion done

    $ ls mymesh*.xml
    mymesh_physical_region.xml  mymesh.xml
    </pre></p>
    Then: Run Dolfin with those XML meshes.
    </article>
  </slide>

  <slide>
    <hgroup>
      <h2>More problems: Time dependence</h2>
    </hgroup>
    <article>
    Your on your own for time-stepping.<br/>
    E.g., backward Euler:
    \[
    \frac{u^{k} - u^{k-1}}{\delta t} = \Delta u^{k} + f
    \]
    \[
    (I-\delta t \Delta) u^{k} = u^{k-1} + (\delta t) f
    \]
    Build weak form, put into FEniCS:
    <p><pre class="prettyprint" data-lang="python">
    a = u*v*dx + dt*inner(grad(u), grad(v))*dx
    L = y_1*v*dx + dt*f*v*dx
    while t &lt; T:
        solve(a == L, y)
        t += dt
        y_1.assign(y)
    </pre></p>
    </article>
  </slide>

  <!--slide>
    <hgroup>
      <h2>Additional ressources</h2>
    </hgroup>
    <article>
    <ul>
    <li>the FEniCS manual/book (700+ pages), <a href="https://launchpad.net/fenics-book">https://launchpad.net/fenics-book</a></li>
    <li>the Dolfin page <a href="https://launchpad.net/dolfin">https://launchpad.net/dolfin</a></li>
    <li>MA477</li>
    </ul>
    </article>
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