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DAG Dependencies

# Parallel examples¶

Note

Performance numbers from IPython.kernel, not newparallel.

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 pidigits.py file, which is available in the docs/examples/newparallel 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 pidigits.py in your current working directory and then doing:

```In [1]: run pidigits.py
```

### Serial calculation¶

For the serial calculation, we will use SymPy 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 pidigits.py: one_digit_freqs() (which calculates how many times each digit occurs) and plot_one_digit_freqs() (which uses Matplotlib to plot the result). Here is an interactive IPython session that uses these functions with SymPy:

```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:

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 pidigits.py:

```def compute_two_digit_freqs(filename):
"""
Read digits of pi from a file and compute the 2 digit frequencies.
"""
d = txt_file_to_digits(filename)
freqs = two_digit_freqs(d)
return freqs

def reduce_freqs(freqlist):
"""
Add up a list of freq counts to get the total counts.
"""
allfreqs = np.zeros_like(freqlist[0])
for f in freqlist:
allfreqs += f
return allfreqs
```

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

1. Use ipcluster to start 15 engines. We used an 8 core (2 quad core CPUs) cluster with hyperthreading enabled which makes the 8 cores looks like 16 (1 controller + 15 engines) in the OS. However, the maximum speedup we can observe is still only 8x.
2. With the 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 8 core cluster, we observe a speedup of 7.7x. This is slightly less than linear scaling (8x) 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 parallelpi.py interactively into 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]: 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']

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.

## 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 docs/examples/newparallel directory of the IPython source. The function price_options() in mcpricer.py implements the basic Monte Carlo pricing algorithm using the NumPy package and is shown here:

```def price_options(S=100.0, K=100.0, sigma=0.25, r=0.05, days=260, paths=10000):
"""
Price European and Asian options using a Monte Carlo method.

Parameters
----------
S : float
The initial price of the stock.
K : float
The strike price of the option.
sigma : float
The volatility of the stock.
r : float
The risk free interest rate.
days : int
The number of days until the option expires.
paths : int
The number of Monte Carlo paths used to price the option.

Returns
-------
A tuple of (E. call, E. put, A. call, A. put) option prices.
"""
import numpy as np
from math import exp,sqrt

h = 1.0/days
const1 = exp((r-0.5*sigma**2)*h)
const2 = sigma*sqrt(h)
stock_price = S*np.ones(paths, dtype='float64')
stock_price_sum = np.zeros(paths, dtype='float64')
for j in range(days):
growth_factor = const1*np.exp(const2*np.random.standard_normal(paths))
stock_price = stock_price*growth_factor
stock_price_sum = stock_price_sum + stock_price
stock_price_avg = stock_price_sum/days
zeros = np.zeros(paths, dtype='float64')
r_factor = exp(-r*h*days)
euro_put = r_factor*np.mean(np.maximum(zeros, K-stock_price))
asian_put = r_factor*np.mean(np.maximum(zeros, K-stock_price_avg))
euro_call = r_factor*np.mean(np.maximum(zeros, stock_price-K))
asian_call = r_factor*np.mean(np.maximum(zeros, stock_price_avg-K))
return (euro_call, euro_put, asian_call, asian_put)
```

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

```#!/usr/bin/env python
"""Run a Monte-Carlo options pricer in parallel."""

#-----------------------------------------------------------------------------
# Imports
#-----------------------------------------------------------------------------

import sys
import time
from IPython.parallel import Client
import numpy as np
from mcpricer import price_options
from matplotlib import pyplot as plt

#-----------------------------------------------------------------------------
# Setup parameters for the run
#-----------------------------------------------------------------------------

s = '%s [%r]: ' % (text, the_type(default))
result = raw_input(s)
if result:
return the_type(result)
else:
return the_type(default)

cluster_profile = ask_question("Cluster profile", str, "default")
price = ask_question("Initial price", float, 100.0)
rate = ask_question("Interest rate", float, 0.05)
days = ask_question("Days to expiration", int, 260)
paths = ask_question("Number of MC paths", int, 10000)
n_strikes = ask_question("Number of strike values", int, 5)
min_strike = ask_question("Min strike price", float, 90.0)
max_strike = ask_question("Max strike price", float, 110.0)
n_sigmas = ask_question("Number of volatility values", int, 5)
min_sigma = ask_question("Min volatility", float, 0.1)
max_sigma = ask_question("Max volatility", float, 0.4)

strike_vals = np.linspace(min_strike, max_strike, n_strikes)
sigma_vals = np.linspace(min_sigma, max_sigma, n_sigmas)

#-----------------------------------------------------------------------------
# Setup for parallel calculation
#-----------------------------------------------------------------------------

# The Client is used to setup the calculation and works with all
# engines.
c = Client(profile=cluster_profile)

# A LoadBalancedView is an interface to the engines that provides dynamic load
# balancing at the expense of not knowing which engine will execute the code.

# Initialize the common code on the engines. This Python module has the
# price_options function that prices the options.

#-----------------------------------------------------------------------------
# Perform parallel calculation
#-----------------------------------------------------------------------------

print "Running parallel calculation over strike prices and volatilities..."
print "Strike prices: ", strike_vals
print "Volatilities: ", sigma_vals
sys.stdout.flush()

t1 = time.time()
async_results = []
for strike in strike_vals:
for sigma in sigma_vals:
ar = view.apply_async(price_options, price, strike, sigma, rate, days, paths)
async_results.append(ar)

sys.stdout.flush()

# Block until all tasks are completed.
c.wait(async_results)
t2 = time.time()
t = t2-t1

print "Parallel calculation completed, time = %s s" % t
print "Collecting results..."

results = [ar.get() for ar in async_results]

# Assemble the result into a structured NumPy array.
prices = np.empty(n_strikes*n_sigmas,
dtype=[('ecall',float),('eput',float),('acall',float),('aput',float)]
)

for i, price in enumerate(results):
prices[i] = tuple(price)

prices.shape = (n_strikes, n_sigmas)
strike_mesh, sigma_mesh = np.meshgrid(strike_vals, sigma_vals)

print "Results are available: strike_mesh, sigma_mesh, prices"
print "To plot results type 'plot_options(sigma_mesh, strike_mesh, prices)'"

#-----------------------------------------------------------------------------
# Utilities
#-----------------------------------------------------------------------------

def plot_options(sigma_mesh, strike_mesh, prices):
"""
Make a contour plot of the option price in (sigma, strike) space.
"""
plt.figure(1)

plt.subplot(221)
plt.contourf(sigma_mesh, strike_mesh, prices['ecall'])
plt.axis('tight')
plt.colorbar()
plt.title('European Call')
plt.ylabel("Strike Price")

plt.subplot(222)
plt.contourf(sigma_mesh, strike_mesh, prices['acall'])
plt.axis('tight')
plt.colorbar()
plt.title("Asian Call")

plt.subplot(223)
plt.contourf(sigma_mesh, strike_mesh, prices['eput'])
plt.axis('tight')
plt.colorbar()
plt.title("European Put")
plt.xlabel("Volatility")
plt.ylabel("Strike Price")

plt.subplot(224)
plt.contourf(sigma_mesh, strike_mesh, prices['aput'])
plt.axis('tight')
plt.colorbar()
plt.title("Asian Put")
plt.xlabel("Volatility")

```

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

```In [7]: run mcdriver.py
Submitted tasks:  [0, 1, 2, ...]
```

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

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

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

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

These results are shown in the two figures below. On a 8 core cluster the entire calculation (10 strike prices, 10 volatilities, 100,000 paths for each) took 30 seconds in parallel, giving a speedup of 7.7x, which is comparable to the speedup observed in our previous example.

## 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 DirectView and 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 Windows HPC Server 2008. IPython’s built in support for the Windows HPC job scheduler makes it easy to get started with IPython’s parallel capabilities.

Note

The newparallel code has never been run on Windows HPC Server, so the last conclusion is untested.