=================
Parallel examples
=================

In this section we describe two more involved examples of using an IPython
cluster to perform a parallel computation. In these examples, we will be using
IPython's "pylab" mode, which enables interactive plotting using the
Matplotlib package. IPython can be started in this mode by typing::

    ipython --pylab

at the system command line.

150 million digits of pi
========================

In this example we would like to study the distribution of digits in the
number pi (in base 10). While it is not known if pi is a normal number (a
number is normal in base 10 if 0-9 occur with equal likelihood) numerical
investigations suggest that it is. We will begin with a serial calculation on
10,000 digits of pi and then perform a parallel calculation involving 150
million digits.

In both the serial and parallel calculation we will be using functions defined
in the :file:`pidigits.py` file, which is available in the
:file:`docs/examples/parallel` directory of the IPython source distribution.
These functions provide basic facilities for working with the digits of pi and
can be loaded into IPython by putting :file:`pidigits.py` in your current
working directory and then doing:

.. sourcecode:: ipython

    In [1]: run pidigits.py

Serial calculation
------------------

For the serial calculation, we will use `SymPy <http://www.sympy.org>`_ to
calculate 10,000 digits of pi and then look at the frequencies of the digits
0-9. Out of 10,000 digits, we expect each digit to occur 1,000 times. While
SymPy is capable of calculating many more digits of pi, our purpose here is to
set the stage for the much larger parallel calculation.

In this example, we use two functions from :file:`pidigits.py`: 
:func:`one_digit_freqs` (which calculates how many times each digit occurs)
and :func:`plot_one_digit_freqs` (which uses Matplotlib to plot the result).
Here is an interactive IPython session that uses these functions with
SymPy:

.. sourcecode:: ipython

    In [7]: import sympy

    In [8]: pi = sympy.pi.evalf(40)

    In [9]: pi
    Out[9]: 3.141592653589793238462643383279502884197

    In [10]: pi = sympy.pi.evalf(10000)

    In [11]: digits = (d for d in str(pi)[2:])  # create a sequence of digits

    In [12]: run pidigits.py  # load one_digit_freqs/plot_one_digit_freqs

    In [13]: freqs = one_digit_freqs(digits)

    In [14]: plot_one_digit_freqs(freqs)
    Out[14]: [<matplotlib.lines.Line2D object at 0x18a55290>]

The resulting plot of the single digit counts shows that each digit occurs
approximately 1,000 times, but that with only 10,000 digits the
statistical fluctuations are still rather large:

.. image:: figs/single_digits.*

It is clear that to reduce the relative fluctuations in the counts, we need
to look at many more digits of pi. That brings us to the parallel calculation.

Parallel calculation
--------------------

Calculating many digits of pi is a challenging computational problem in itself. 
Because we want to focus on the distribution of digits in this example, we
will use pre-computed digit of pi from the website of Professor Yasumasa
Kanada at the University of Tokyo (http://www.super-computing.org). These
digits come in a set of text files (ftp://pi.super-computing.org/.2/pi200m/)
that each have 10 million digits of pi.

For the parallel calculation, we have copied these files to the local hard
drives of the compute nodes. A total of 15 of these files will be used, for a
total of 150 million digits of pi. To make things a little more interesting we
will calculate the frequencies of all 2 digits sequences (00-99) and then plot
the result using a 2D matrix in Matplotlib.

The overall idea of the calculation is simple: each IPython engine will
compute the two digit counts for the digits in a single file. Then in a final
step the counts from each engine will be added up. To perform this
calculation, we will need two top-level functions from :file:`pidigits.py`:

.. literalinclude:: ../../examples/parallel/pi/pidigits.py
   :language: python
   :lines: 47-62

We will also use the :func:`plot_two_digit_freqs` function to plot the
results. The code to run this calculation in parallel is contained in
:file:`docs/examples/parallel/parallelpi.py`. This code can be run in parallel
using IPython by following these steps:

1. Use :command:`ipcluster` to start 15 engines. We used 16 cores of an SGE linux
   cluster (1 controller + 15 engines).
2. With the file :file:`parallelpi.py` in your current working directory, open
   up IPython in pylab mode and type ``run parallelpi.py``.  This will download
   the pi files via ftp the first time you run it, if they are not
   present in the Engines' working directory.

When run on our 16 cores, we observe a speedup of 14.2x. This is slightly
less than linear scaling (16x) because the controller is also running on one of
the cores.

To emphasize the interactive nature of IPython, we now show how the
calculation can also be run by simply typing the commands from
:file:`parallelpi.py` interactively into IPython:

.. sourcecode:: ipython

    In [1]: from IPython.parallel import Client

    # The Client allows us to use the engines interactively.
    # We simply pass Client the name of the cluster profile we
    # are using.
    In [2]: c = Client(profile='mycluster')
    In [3]: v = c[:]

    In [3]: c.ids
    Out[3]: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]

    In [4]: run pidigits.py

    In [5]: filestring = 'pi200m.ascii.%(i)02dof20'

    # Create the list of files to process.
    In [6]: files = [filestring % {'i':i} for i in range(1,16)]

    In [7]: files
    Out[7]: 
    ['pi200m.ascii.01of20',
     'pi200m.ascii.02of20',
     'pi200m.ascii.03of20',
     'pi200m.ascii.04of20',
     'pi200m.ascii.05of20',
     'pi200m.ascii.06of20',
     'pi200m.ascii.07of20',
     'pi200m.ascii.08of20',
     'pi200m.ascii.09of20',
     'pi200m.ascii.10of20',
     'pi200m.ascii.11of20',
     'pi200m.ascii.12of20',
     'pi200m.ascii.13of20',
     'pi200m.ascii.14of20',
     'pi200m.ascii.15of20']
    
    # download the data files if they don't already exist:
    In [8]: v.map(fetch_pi_file, files)
    
    # This is the parallel calculation using the Client.map method 
    # which applies compute_two_digit_freqs to each file in files in parallel.
    In [9]: freqs_all = v.map(compute_two_digit_freqs, files)

    # Add up the frequencies from each engine.
    In [10]: freqs = reduce_freqs(freqs_all)

    In [11]: plot_two_digit_freqs(freqs)
    Out[11]: <matplotlib.image.AxesImage object at 0x18beb110>

    In [12]: plt.title('2 digit counts of 150m digits of pi')
    Out[12]: <matplotlib.text.Text object at 0x18d1f9b0>

The resulting plot generated by Matplotlib is shown below. The colors indicate
which two digit sequences are more (red) or less (blue) likely to occur in the
first 150 million digits of pi. We clearly see that the sequence "41" is 
most likely and that "06" and "07" are least likely. Further analysis would
show that the relative size of the statistical fluctuations have decreased
compared to the 10,000 digit calculation.

.. image:: figs/two_digit_counts.*


Parallel options pricing
========================

An option is a financial contract that gives the buyer of the contract the
right to buy (a "call") or sell (a "put") a secondary asset (a stock for
example) at a particular date in the future (the expiration date) for a
pre-agreed upon price (the strike price). For this right, the buyer pays the
seller a premium (the option price). There are a wide variety of flavors of
options (American, European, Asian, etc.) that are useful for different
purposes: hedging against risk, speculation, etc.

Much of modern finance is driven by the need to price these contracts
accurately based on what is known about the properties (such as volatility) of
the underlying asset. One method of pricing options is to use a Monte Carlo
simulation of the underlying asset price. In this example we use this approach
to price both European and Asian (path dependent) options for various strike
prices and volatilities.

The code for this example can be found in the :file:`docs/examples/parallel/options`
directory of the IPython source. The function :func:`price_options` in
:file:`mckernel.py` implements the basic Monte Carlo pricing algorithm using
the NumPy package and is shown here:

.. literalinclude:: ../../examples/parallel/options/mckernel.py
   :language: python

To run this code in parallel, we will use IPython's :class:`LoadBalancedView` class,
which distributes work to the engines using dynamic load balancing. This
view is a wrapper of the :class:`Client` class shown in
the previous example. The parallel calculation using :class:`LoadBalancedView` can
be found in the file :file:`mcpricer.py`. The code in this file creates a
:class:`LoadBalancedView` instance and then submits a set of tasks using
:meth:`LoadBalancedView.apply` that calculate the option prices for different
volatilities and strike prices. The results are then plotted as a 2D contour
plot using Matplotlib.

.. literalinclude:: ../../examples/parallel/options/mcpricer.py
   :language: python

To use this code, start an IPython cluster using :command:`ipcluster`, open
IPython in the pylab mode with the file :file:`mckernel.py` in your current
working directory and then type:

.. sourcecode:: ipython

    In [7]: run mcpricer.py
    
    Submitted tasks:  30

Once all the tasks have finished, the results can be plotted using the
:func:`plot_options` function. Here we make contour plots of the Asian
call and Asian put options as function of the volatility and strike price:

.. sourcecode:: ipython

    In [8]: plot_options(sigma_vals, strike_vals, prices['acall'])

    In [9]: plt.figure()
    Out[9]: <matplotlib.figure.Figure object at 0x18c178d0>

    In [10]: plot_options(sigma_vals, strike_vals, prices['aput'])

These results are shown in the two figures below. On our 15 engines, the
entire calculation (15 strike prices, 15 volatilities, 100,000 paths for each)
took 37 seconds in parallel, giving a speedup of 14.1x, which is comparable
to the speedup observed in our previous example.

.. image:: figs/asian_call.*

.. image:: figs/asian_put.*

Conclusion
==========

To conclude these examples, we summarize the key features of IPython's
parallel architecture that have been demonstrated:

* Serial code can be parallelized often with only a few extra lines of code.
  We have used the :class:`DirectView` and :class:`LoadBalancedView` classes
  for this purpose.
* The resulting parallel code can be run without ever leaving the IPython's
  interactive shell.
* Any data computed in parallel can be explored interactively through 
  visualization or further numerical calculations.
* We have run these examples on a cluster running RHEL 5 and Sun GridEngine.
  IPython's built in support for SGE (and other batch systems) makes it easy
  to get started with IPython's parallel capabilities.

