$$ \newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\tp}{\thinspace .} $$

Study Guide: Scientific software engineering for a simple ODE problem

Hans Petter Langtangen [1, 2]

[1] Center for Biomedical Computing, Simula Research Laboratory
[2] Department of Informatics, University of Oslo

Sep 25, 2014






Table of contents

Creating user interfaces
      Accessing command-line arguments
      Reading a sequence of command-line arguments
      Implementation
      Working with an argument parser
      Reading option-values pairs
      A graphical user interface
      The Parampool package
      Making a compute function
      The hard part of the compute function: the HTML code
      How to embed a PNG plot in HTML code
      Generating the user interface
      Running the web application
      More advanced use
Computing convergence rates
      Estimating the convergence rate \( r \)
      Implementation
      Execution
      Debugging via convergence rates
Software engineering
      Making a module
      Sketch of the module
      Test block
      Extended test block
      Prefixing imported functions by the module name
      Downside of module prefix notation
      Doctests
      Running doctests
      Unit testing with nose
      Basic use of nose
      Example on a nose test in the source code
      Example on a nose test in a separate file
      The habit of writing nose tests
      Purpose of a test function: raise AssertionError if failure
      Advantages of nose
      Demonstrating nose (ideas)
      Demonstrating nose (code)
      Floats as test results require careful comparison
      Test of wrong use
      Test of convergence rates
      Classical unit testing with unittest
      Basic use of unittest
      Demonstration of unittest
Implementing simple problem and solver classes
      What to learn
      The problem class
      Improved problem class
      The solver class
      The visualizer class
      Combing the classes
Implementing more advanced problem and solver classes
      A generic class for parameters
      The problem class
      The solver class
      The visualizer class
Performing scientific experiments
      Model problem and numerical solution method
      Plan for the experiments
      Typical plot summarizing the results
      Script code
      Comments to the code
      Interpreting output from other programs
      Code for grabbing output from another program
      Code for interpreting the grabbed output
      Making a report
      Publishing a complete project






Creating user interfaces






Accessing command-line arguments 

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Terminal> python myprog.py 1.5 2 0.5 0.8 0.4
Terminal> python myprog.py --I 1.5 --a 2 --dt 0.8 0.4






Reading a sequence of command-line arguments

The program decay_plot.py needs this input:

Give these on the command line in correct sequence 

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Terminal> python decay_cml.py 1.5 2 0.5 0.8 0.4






Implementation 

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import sys

def read_command_line():
    if len(sys.argv) < 6:
        print 'Usage: %s I a T on/off dt1 dt2 dt3 ...' % \ 
              sys.argv[0]; sys.exit(1)  # abort

    I = float(sys.argv[1])
    a = float(sys.argv[2])
    T = float(sys.argv[3])
    makeplot = sys.argv[4] in ('on', 'True')
    dt_values = [float(arg) for arg in sys.argv[5:]]

    return I, a, T, makeplot, dt_values

Note:

Complete program: decay_cml.py.






Working with an argument parser

Set option-value pairs on the command line if the default value is not suitable: 

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Terminal> python decay_argparse.py --I 1.5 --a 2 --dt 0.8 0.4

Code: 

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def define_command_line_options():
    import argparse
    parser = argparse.ArgumentParser()
    parser.add_argument('--I', '--initial_condition', type=float,
                        default=1.0, help='initial condition, u(0)',
                        metavar='I')
    parser.add_argument('--a', type=float,
                        default=1.0, help='coefficient in ODE',
                        metavar='a')
    parser.add_argument('--T', '--stop_time', type=float,
                        default=1.0, help='end time of simulation',
                        metavar='T')
    parser.add_argument('--makeplot', action='store_true',
                        help='display plot or not')
    parser.add_argument('--dt', '--time_step_values', type=float,
                        default=[1.0], help='time step values',
                        metavar='dt', nargs='+', dest='dt_values')
    return parser

(metavar is the symbol used in help output)






Reading option-values pairs

argparse.ArgumentParser parses the command-line arguments: 

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def read_command_line():
    parser = define_command_line_options()
    args = parser.parse_args()
    print 'I={}, a={}, T={}, makeplot={}, dt_values={}'.format(
        args.I, args.a, args.T, args.makeplot, args.dt_values)
    return args.I, args.a, args.T, args.makeplot, args.dt_values

Complete program: decay_argparse.py.






A graphical user interface

Normally very much programming required - and much competence on graphical user interfaces.

Here: use a tool to automatically create it in a few minutes (!)






The Parampool package

Remark.

The forthcoming material aims at those with particular interest in equipping their programs with a GUI - others can safely skip it.






Making a compute function

The compute function main_GUI: 

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def main_GUI(I=1.0, a=.2, T=4.0,
         dt_values=[1.25, 0.75, 0.5, 0.1],
         theta_values=[0, 0.5, 1]):






The hard part of the compute function: the HTML code

Suppose explore solves the problem, makes a plot, computes the error and returns appropriate HTML code with the plot. Embed error and plots in a table: 

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def main_GUI(I=1.0, a=.2, T=4.0,
         dt_values=[1.25, 0.75, 0.5, 0.1],
         theta_values=[0, 0.5, 1]):
    # Build HTML code for web page. Arrange plots in columns
    # corresponding to the theta values, with dt down the rows
    theta2name = {0: 'FE', 1: 'BE', 0.5: 'CN'}
    html_text = '<table>\n'
    for dt in dt_values:
        html_text += '<tr>\n'
        for theta in theta_values:
            E, html = explore(I, a, T, dt, theta, makeplot=True)
            html_text += """
<td>
<center><b>%s, dt=%g, error: %s</b></center><br>
%s
</td>
""" % (theta2name[theta], dt, E, html)
        html_text += '</tr>\n'
    html_text += '</table>\n'
    return html_text






How to embed a PNG plot in HTML code

In explore: 

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import matplotlib.pyplot as plt
...
# plot
plt.plot(t, u, r-')
plt.xlabel('t')
plt.ylabel('u')
...
from parampool.utils import save_png_to_str
html_text = save_png_to_str(plt, plotwidth=400)

If you know HTML, you can return more sophisticated layout etc.






Generating the user interface

Make a file decay_GUI_generate.py: 

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from parampool.generator.flask import generate
from decay_GUI import main
generate(main,
         output_controller='decay_GUI_controller.py',
         output_template='decay_GUI_view.py',
         output_model='decay_GUI_model.py')

Running decay_GUI_generate.py results in

  1. decay_GUI_model.py defines HTML widgets to be used to set input data in the web interface,

  2. templates/decay_GUI_views.py defines the layout of the web page,

  3. decay_GUI_controller.py runs the web application.
Good news: we only need to run decay_GUI_controller.py and there is no need to look into any of these files!






Running the web application

Start the GUI 

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Terminal> python decay_GUI_controller.py

Open a web browser at 127.0.0.1:5000






More advanced use






Computing convergence rates

Frequent assumption on the relation between the numerical error \( E \) and some discretization parameter \( \Delta t \): $$ \begin{equation} E = C\Delta t^r, \label{decay:E:dt} \end{equation} $$






Estimating the convergence rate \( r \)

Perform numerical experiments: \( (\Delta t_i, E_i) \), \( i=0,\ldots,m-1 \). Two methods for finding \( r \) (and \( C \)):

  1. Take the logarithm of \eqref{decay:E:dt}, \( \ln E = r\ln \Delta t + \ln C \), and fit a straight line to the data points \( (\Delta t_i, E_i) \), \( i=0,\ldots,m-1 \).

  2. Consider two consecutive experiments, \( (\Delta t_i, E_i) \) and \( (\Delta t_{i-1}, E_{i-1}) \). Dividing the equation \( E_{i-1}=C\Delta t_{i-1}^r \) by \( E_{i}=C\Delta t_{i}^r \) and solving for \( r \) yields
$$ \begin{equation} r_{i-1} = \frac{\ln (E_{i-1}/E_i)}{\ln (\Delta t_{i-1}/\Delta t_i)} \label{decay:conv:rate} \end{equation} $$ for \( i=1,=\ldots,m-1 \).

Method 2 is best.






Implementation

Compute \( r_0, r_1, \ldots, r_{m-2} \): 

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from math import log

def main():
    I, a, T, makeplot, dt_values = read_command_line()
    r = {}  # estimated convergence rates
    for theta in 0, 0.5, 1:
        E_values = []
        for dt in dt_values:
            E = explore(I, a, T, dt, theta, makeplot=False)
            E_values.append(E)

        # Compute convergence rates
        m = len(dt_values)
        r[theta] = [log(E_values[i-1]/E_values[i])/
                    log(dt_values[i-1]/dt_values[i])
                    for i in range(1, m, 1)]

    for theta in r:
        print '\nPairwise convergence rates for theta=%g:' % theta
        print ' '.join(['%.2f' % r_ for r_ in r[theta]])
    return r

Complete program: decay_convrate.py.






Execution 

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Terminal> python decay_convrate.py --dt 0.5 0.25 0.1 0.05 0.025 0.01
...
Pairwise convergence rates for theta=0:
1.33 1.15 1.07 1.03 1.02

Pairwise convergence rates for theta=0.5:
2.14 2.07 2.03 2.01 2.01

Pairwise convergence rates for theta=1:
0.98 0.99 0.99 1.00 1.00

Strong verification method.

Verify that \( r \) has the expected value!






Debugging via convergence rates

Potential bug: missing a in the denominator, 

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u[n+1] = (1 - (1-theta)*a*dt)/(1 + theta*dt)*u[n]

Running decay_convrate.py gives same rates.

Why? The value of \( a \)... (\( a=1 \))

0 and 1 are bad values in tests!

Better: 

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Terminal> python decay_convrate.py --a 2.1 --I 0.1  \ 
          --dt 0.5 0.25 0.1 0.05 0.025 0.01
...
Pairwise convergence rates for theta=0:
1.49 1.18 1.07 1.04 1.02

Pairwise convergence rates for theta=0.5:
-1.42 -0.22 -0.07 -0.03 -0.01

Pairwise convergence rates for theta=1:
0.21 0.12 0.06 0.03 0.01

Forward Euler works...because \( \theta=0 \) hides the bug.

This bug gives \( r\approx 0 \): 

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u[n+1] = ((1-theta)*a*dt)/(1 + theta*dt*a)*u[n]






Software engineering

Goal: make more professional numerical software.

Topics:






Making a module

Module name: decay_mod, filename: decay_mod.py.






Sketch of the module 

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from numpy import *
from matplotlib.pyplot import *
import sys

def solver(I, a, T, dt, theta):
    ...

def verify_three_steps():
    ...

def verify_exact_discrete_solution():
    ...

def u_exact(t, I, a):
    ...

def explore(I, a, T, dt, theta=0.5, makeplot=True):
    ...

def define_command_line_options():
    ...

def read_command_line(use_argparse=True):
    ...

def main():
    ...

That is! It's a module decay_mod in file decay_mod.py.

Usage in some other program: 

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from decay_mod import solver
u, t = solver(I=1.0, a=3.0, T=3, dt=0.01, theta=0.5)






Test block

At the end of a module it is common to include a test block: 

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if __name__ == '__main__':
    main()

Note:






Extended test block 

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if __name__ == '__main__':
    if 'verify' in sys.argv:
        if verify_three_steps() and verify_discrete_solution():
            pass # ok
        else:
            print 'Bug in the implementation!'
    elif 'verify_rates' in sys.argv:
        sys.argv.remove('verify_rates')
        if not '--dt' in sys.argv:
            print 'Must assign several dt values'
            sys.exit(1)  # abort
        if verify_convergence_rate():
            pass
        else:
            print 'Bug in the implementation!'
    else:
        # Perform simulations
        main()






Prefixing imported functions by the module name 

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from numpy import *
from matplotlib.pyplot import *

This imports a large number of names (sin, exp, linspace, plot, ...).

Confusion: is a function from numpy? Or matplotlib.pyplot?

Alternative (recommended) import: 

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import numpy
import matplotlib.pyplot

Now we need to prefix functions with module name: 

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t = numpy.linspace(0, T, Nt+1)
u_e = I*numpy.exp(-a*t)
matplotlib.pyplot.plot(t, u_e)

Common standard: 

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import numpy as np
import matplotlib.pyplot as plt

t = np.linspace(0, T, Nt+1)
u_e = I*np.exp(-a*t)
plt.plot(t, u_e)






Downside of module prefix notation

A math line like \( e^{-at}\sin(2\pi t) \) gets cluttered with module names, 

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numpy.exp(-a*t)*numpy.sin(2(numpy.pi*t)
# or
np.exp(-a*t)*np.sin(2*np.pi*t)

Solution (much used in this course): do two imports 

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import numpy as np
from numpy import exp, sin, pi
...
t = np.linspace(0, T, Nt+1)
u_e = exp(-a*t)*sin(2*pi*t)






Doctests

Doc strings can be equipped with interactive Python sessions for demonstrating usage and automatic testing of functions. 

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def solver(I, a, T, dt, theta):
    """
    Solve u'=-a*u, u(0)=I, for t in (0,T] with steps of dt.


    >>> u, t = solver(I=0.8, a=1.2, T=4, dt=0.5, theta=0.5)
    >>> for t_n, u_n in zip(t, u):
    ...     print 't=%.1f, u=%.14f' % (t_n, u_n)
    t=0.0, u=0.80000000000000
    t=0.5, u=0.43076923076923
    t=1.0, u=0.23195266272189
    t=1.5, u=0.12489758761948
    t=2.0, u=0.06725254717972
    t=2.5, u=0.03621291001985
    t=3.0, u=0.01949925924146
    t=3.5, u=0.01049960113002
    t=4.0, u=0.00565363137770
    """
    ...






Running doctests

Automatic check that the code reproduces the doctest output: 

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Terminal> python -m doctest decay_mod_doctest.py

Report in case of failure: 

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Terminal> python -m doctest decay_mod_doctest.py
********************************************************
File "decay_mod_doctest.py", line 12, in decay_mod_doctest....
Failed example:
    for t_n, u_n in zip(t, u):
        print 't=%.1f, u=%.14f' % (t_n, u_n)
Expected:
    t=0.0, u=0.80000000000000
    t=0.5, u=0.43076923076923
    t=1.0, u=0.23195266272189
    t=1.5, u=0.12489758761948
    t=2.0, u=0.06725254717972
Got:
    t=0.0, u=0.80000000000000
    t=0.5, u=0.43076923076923
    t=1.0, u=0.23195266272189
    t=1.5, u=0.12489758761948
    t=2.0, u=0.06725254718756
********************************************************
1 items had failures:
   1 of   2 in decay_mod_doctest.solver
***Test Failed*** 1 failures.

Floats are difficult to compare.

Limit the number of digits in the output in doctests! Otherwise, round-off errors on a different machine may ruin the test.

Complete program: decay_mod_doctest.py.






Unit testing with nose






Basic use of nose

  1. Implement tests in test functions with names starting with test_.

  2. Test functions cannot have arguments.

  3. Test functions perform assertions on computed results using assert functions from the nose.tools module.

  4. Test functions can be in the source code files or be collected in separate files test*.py.





Example on a nose test in the source code

Very simple module mymod (in file mymod.py): 

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def double(n):
    return 2*n

Write test function in mymod.py: 

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def double(n):
    return 2*n

import nose.tools as nt

def test_double():
    result = double(4)
    nt.assert_equal(result, 8)

Running 

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Terminal> nosetests -s mymod

makes the nose tool run all test_*() functions in mymod.py.






Example on a nose test in a separate file

Write the test in a separate file, say test_mymod.py: 

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import nose.tools as nt
import mymod

def test_double():
    result = mymod.double(4)
    nt.assert_equal(result, 8)

Running 

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Terminal> nosetests -s

makes the nose tool run all test_*() functions in all files test*.py in the current directory and in all subdirectories (recursevely) with names tests or *_tests.

Tip.

Start with test functions in the source code file. When the file contains many tests, or when you have many source code files, move tests to separate files.






The habit of writing nose tests






Purpose of a test function: raise AssertionError if failure

Alternative ways of raising AssertionError if result is not 8: 

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import nose.tools as nt

def test_double():
    result = ...

    nt.assert_equal(result, 8)    # alternative 1

    assert result == 8            # alternative 2

    if result != 8:               # alternative 3
        raise AssertionError()






Advantages of nose






Demonstrating nose (ideas)

Aim: test function solver for \( u'=-au \), \( u(0)=I \).

We design three unit tests:

  1. A comparison between the computed \( u^n \) values and the exact discrete solution

  2. A comparison between the computed \( u^n \) values and precomputed verified reference values

  3. A comparison between observed and expected convergence rates
These tests follow very closely the previous verify* functions.






Demonstrating nose (code) 

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import nose.tools as nt
import decay_mod_unittest as decay_mod
import numpy as np

def exact_discrete_solution(n, I, a, theta, dt):
    """Return exact discrete solution of the theta scheme."""
    dt = float(dt)  # avoid integer division
    factor = (1 - (1-theta)*a*dt)/(1 + theta*dt*a)
    return I*factor**n

def test_exact_discrete_solution():
    """
    Compare result from solver against
    formula for the discrete solution.
    """
    theta = 0.8; a = 2; I = 0.1; dt = 0.8
    N = int(8/dt)  # no of steps
    u, t = decay_mod.solver(I=I, a=a, T=N*dt, dt=dt, theta=theta)
    u_de = np.array([exact_discrete_solution(n, I, a, theta, dt)
                     for n in range(N+1)])
    diff = np.abs(u_de - u).max()
    nt.assert_almost_equal(diff, 0, delta=1E-14)






Floats as test results require careful comparison 

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def test_solver():
    """
    Compare result from solver against
    precomputed arrays for theta=0, 0.5, 1.
    """
    I=0.8; a=1.2; T=4; dt=0.5  # fixed parameters
    precomputed = {
        't': np.array([ 0. ,  0.5,  1. ,  1.5,  2. ,  2.5,
                        3. ,  3.5,  4. ]),
        0.5: np.array(
            [ 0.8       ,  0.43076923,  0.23195266, 0.12489759,
              0.06725255,  0.03621291,  0.01949926, 0.0104996 ,
              0.00565363]),
        0: ...,
        1: ...
        }
    for theta in 0, 0.5, 1:
        u, t = decay_mod.solver(I, a, T, dt, theta=theta)
        diff = np.abs(u - precomputed[theta]).max()
        # Precomputed numbers are known to 8 decimal places
        nt.assert_almost_equal(diff, 0, places=8,
                               msg='theta=%s' % theta)






Test of wrong use

Example: 

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theta = 1; a = 1; I = 1; dt = 2

may lead to integer division: 

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(1 - (1-theta)*a*dt)  # becomes 1
(1 + theta*dt*a)      # becomes 2
(1 - (1-theta)*a*dt)/(1 + theta*dt*a)  # becomes 0 (!)

Test that solver does not suffer from such integer division: 

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def test_potential_integer_division():
    """Choose variables that can trigger integer division."""
    theta = 1; a = 1; I = 1; dt = 2
    N = 4
    u, t = decay_mod.solver(I=I, a=a, T=N*dt, dt=dt, theta=theta)
    u_de = np.array([exact_discrete_solution(n, I, a, theta, dt)
                     for n in range(N+1)])
    diff = np.abs(u_de - u).max()
    nt.assert_almost_equal(diff, 0, delta=1E-14)






Test of convergence rates

Convergence rate tests are very common for differential equation solvers. 

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def test_convergence_rates():
    """Compare empirical convergence rates to exact ones."""
    # Set command-line arguments directly in sys.argv
    import sys
    sys.argv[1:] = '--I 0.8 --a 2.1 --T 5 '\ 
                   '--dt 0.4 0.2 0.1 0.05 0.025'.split()
    r = decay_mod.main()
    for theta in r:
        nt.assert_true(r[theta])  # check for non-empty list

    expected_rates = {0: 1, 1: 1, 0.5: 2}
    for theta in r:
        r_final = r[theta][-1]
        # Compare to 1 decimal place
        nt.assert_almost_equal(expected_rates[theta], r_final,
                               places=1, msg='theta=%s' % theta)

Complete program: test_decay_nose.py.






Classical unit testing with unittest

Remark.

You will probably not use it, but you're not educated unless you know what unit testing with classes is.






Basic use of unittest

Write file test_mymod.py: 

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import unittest
import mymod

class TestMyCode(unittest.TestCase):
    def test_double(self):
        result = mymod.double(4)
        self.assertEqual(result, 8)

if __name__ == '__main__':
    unittest.main()






Demonstration of unittest 

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import unittest
import decay_mod_unittest as decay
import numpy as np

def exact_discrete_solution(n, I, a, theta, dt):
    factor = (1 - (1-theta)*a*dt)/(1 + theta*dt*a)
    return I*factor**n

class TestDecay(unittest.TestCase):

    def test_exact_discrete_solution(self):
        ...
        diff = np.abs(u_de - u).max()
        self.assertAlmostEqual(diff, 0, delta=1E-14)

    def test_solver(self):
        ...
        for theta in 0, 0.5, 1:
            ...
            self.assertAlmostEqual(diff, 0, places=8,
                                   msg='theta=%s' % theta)

    def test_potential_integer_division():
        ...
        self.assertAlmostEqual(diff, 0, delta=1E-14)

    def test_convergence_rates(self):
        ...
        for theta in r:
            ...
            self.assertAlmostEqual(...)

if __name__ == '__main__':
    unittest.main()

Complete program: test_decay_unittest.py.






Implementing simple problem and solver classes






What to learn

Tasks:

Ideas:




The problem class

Implementation: 

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from numpy import exp

class Problem:
    def __init__(self, I=1, a=1, T=10):
        self.T, self.I, self.a = I, float(a), T

    def u_exact(self, t):
        I, a = self.I, self.a     # extract local variables
        return I*exp(-a*t)

Basic usage: 

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problem = Problem(T=5)
problem.T = 8
problem.dt = 1.5






Improved problem class

More flexible input from the command line: 

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class Problem:
    def __init__(self, I=1, a=1, T=10):
        self.T, self.I, self.a = I, float(a), T

    def define_command_line_options(self, parser=None):
        if parser is None:
            import argparse
            parser = argparse.ArgumentParser()

        parser.add_argument(
            '--I', '--initial_condition', type=float,
            default=self.I, help='initial condition, u(0)',
            metavar='I')
        parser.add_argument(
            '--a', type=float, default=self.a,
            help='coefficient in ODE', metavar='a')
        parser.add_argument(
            '--T', '--stop_time', type=float, default=self.T,
            help='end time of simulation', metavar='T')
        return parser

    def init_from_command_line(self, args):
        self.I, self.a, self.T = args.I, args.a, args.T

    def exact_solution(self, t):
        I, a = self.I, self.a
        return I*exp(-a*t)






The solver class

Implementation: 

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class Solver:
    def __init__(self, problem, dt=0.1, theta=0.5):
        self.problem = problem
        self.dt, self.theta = float(dt), theta

    def define_command_line_options(self, parser):
        parser.add_argument(
            '--dt', '--time_step_value', type=float,
            default=0.5, help='time step value', metavar='dt')
        parser.add_argument(
            '--theta', type=float, default=0.5,
            help='time discretization parameter', metavar='dt')
        return parser

    def init_from_command_line(self, args):
        self.dt, self.theta = args.dt, args.theta

    def solve(self):
        from decay_mod import solver
        self.u, self.t = solver(
            self.problem.I, self.problem.a, self.problem.T,
            self.dt, self.theta)

Note: reuse of the numerical algorithm from the decay_mod module (i.e., the class is a wrapper of the procedural implementation).






The visualizer class 

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class Visualizer:
    def __init__(self, problem, solver):
        self.problem, self.solver = problem, solver

    def plot(self, include_exact=True, plt=None):
        """
        Add solver.u curve to the plotting object plt,
        and include the exact solution if include_exact is True.
        This plot function can be called several times (if
        the solver object has computed new solutions).
        """
        if plt is None:
            import scitools.std  as plt # can use matplotlib as well

        plt.plot(self.solver.t, self.solver.u, '--o')
        plt.hold('on')
        theta2name = {0: 'FE', 1: 'BE', 0.5: 'CN'}
        name = theta2name.get(self.solver.theta, '')
        legends = ['numerical %s' % name]
        if include_exact:
            t_e = linspace(0, self.problem.T, 1001)
            u_e = self.problem.exact_solution(t_e)
            plt.plot(t_e, u_e, 'b-')
            legends.append('exact')
        plt.legend(legends)
        plt.xlabel('t')
        plt.ylabel('u')
        plt.title('theta=%g, dt=%g' %
                  (self.solver.theta, self.solver.dt))
        plt.savefig('%s_%g.png' % (name, self.solver.dt))
        return plt

Remark: The plt object in plot adds a new curve to a plot, which enables comparing different solutions from different runs of Solver.solve






Combing the classes

Let Problem, Solver, and Visualizer play together: 

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def main():
    problem = Problem()
    solver = Solver(problem)
    viz = Visualizer(problem, solver)

    # Read input from the command line
    parser = problem.define_command_line_options()
    parser = solver. define_command_line_options(parser)
    args = parser.parse_args()
    problem.init_from_command_line(args)
    solver. init_from_command_line(args)

    # Solve and plot
    solver.solve()
    import matplotlib.pyplot as plt
    #import scitools.std as plt
    plt = viz.plot(plt=plt)
    E = solver.error()
    if E is not None:
        print 'Error: %.4E' % E
    plt.show()

Complete program: decay_class.py.






Implementing more advanced problem and solver classes






A generic class for parameters 

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class Parameters:
    def set(self, **parameters):
        for name in parameters:
            self.prms[name] = parameters[name]

    def get(self, name):
        return self.prms[name]

    def define_command_line_options(self, parser=None):
        if parser is None:
            import argparse
            parser = argparse.ArgumentParser()

        for name in self.prms:
            tp = self.types[name] if name in self.types else str
            help = self.help[name] if name in self.help else None
            parser.add_argument(
                '--' + name, default=self.get(name), metavar=name,
                type=tp, help=help)

        return parser

    def init_from_command_line(self, args):
        for name in self.prms:
            self.prms[name] = getattr(args, name)

Slightly more advanced version in class_decay_verf1.py.






The problem class 

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class Problem(Parameters):
    """
    Physical parameters for the problem u'=-a*u, u(0)=I,
    with t in [0,T].
    """
    def __init__(self):
        self.prms = dict(I=1, a=1, T=10)
        self.types = dict(I=float, a=float, T=float)
        self.help = dict(I='initial condition, u(0)',
                         a='coefficient in ODE',
                         T='end time of simulation')

    def exact_solution(self, t):
        I, a = self.get('I'), self.get('a')
        return I*np.exp(-a*t)






The solver class 

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class Solver(Parameters):
    def __init__(self, problem):
        self.problem = problem
        self.prms = dict(dt=0.5, theta=0.5)
        self.types = dict(dt=float, theta=float)
        self.help = dict(dt='time step value',
                         theta='time discretization parameter')

    def solve(self):
        from decay_mod import solver
        self.u, self.t = solver(
            self.problem.get('I'),
            self.problem.get('a'),
            self.problem.get('T'),
            self.get('dt'),
            self.get('theta'))

    def error(self):
        try:
            u_e = self.problem.exact_solution(self.t)
            e = u_e - self.u
            E = np.sqrt(self.get('dt')*np.sum(e**2))
        except AttributeError:
            E = None
        return E






The visualizer class






Performing scientific experiments

Goal: explore the behavior of a numerical method for a differential equation and show how scientific experiments can be set up and reported.

Tasks:

Tools to learn:




Model problem and numerical solution method

Problem: $$ \begin{equation} u'(t) = -au(t),\quad u(0)=I,\ 0 < t \leq T, \label{decay:experiments:model} \end{equation} $$

Solution method (\( \theta \)-rule): $$ u^{n+1} = \frac{1 - (1-\theta) a\Delta t}{1 + \theta a\Delta t}u^n, \quad u^0=I\tp $$






Plan for the experiments






Typical plot summarizing the results






Script code

Typical script (small administering program) for running the experiments: 

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import os, sys

def run_experiments(I=1, a=2, T=5):
    # The command line must contain dt values
    if len(sys.argv) > 1:
        dt_values = [float(arg) for arg in sys.argv[1:]]
    else:
        print 'Usage: %s dt1 dt2 dt3 ...' %  sys.argv[0]
        sys.exit(1)  # abort

    # Run module file as a stand-alone application
    cmd = 'python decay_mod.py --I %g --a %g --makeplot --T %g' % \
          (I, a, T)
    dt_values_str = ' '.join([str(v) for v in dt_values])
    cmd += ' --dt %s' % dt_values_str
    print cmd
    failure = os.system(cmd)
    if failure:
        print 'Command failed:', cmd; sys.exit(1)

    # Combine images into rows with 2 plots in each row
    image_commands = []
    for method in 'BE', 'CN', 'FE':
        pdf_files = ' '.join(['%s_%g.pdf' % (method, dt)
                              for dt in dt_values])
        png_files = ' '.join(['%s_%g.png' % (method, dt)
                              for dt in dt_values])
        image_commands.append(
            'montage -background white -geometry 100%' +
            ' -tile 2x %s %s.png' % (png_files, method))
        image_commands.append(
            'convert -trim %s.png %s.png' % (method, method))
        image_commands.append(
            'convert %s.png -transparent white %s.png' %
            (method, method))
        image_commands.append(
            'pdftk %s output tmp.pdf' % pdf_files)
        num_rows = int(round(len(dt_values)/2.0))
        image_commands.append(
            'pdfnup --nup 2x%d tmp.pdf' % num_rows)
        image_commands.append(
            'pdfcrop tmp-nup.pdf %s.pdf' % method)

    for cmd in image_commands:
        print cmd
        failure = os.system(cmd)
        if failure:
            print 'Command failed:', cmd; sys.exit(1)

    # Remove the files generated above and by decay_mod.py
    from glob import glob
    filenames = glob('*_*.png') + glob('*_*.pdf') + \
                glob('*_*.eps') + glob('tmp*.pdf')
    for filename in filenames:
        os.remove(filename)

if __name__ == '__main__':
    run_experiments()

Complete program: experiments/decay_exper0.py.






Comments to the code

Many useful constructs in the previous script:






Interpreting output from other programs

In decay_exper0.py we run a program (os.system) and want to grab the output, e.g., 

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Terminal> python decay_plot_mpl.py
0.0   0.40:    2.105E-01
0.0   0.04:    1.449E-02
0.5   0.40:    3.362E-02
0.5   0.04:    1.887E-04
1.0   0.40:    1.030E-01
1.0   0.04:    1.382E-02

Tasks:






Code for grabbing output from another program

Use the subprocess module to grab output: 

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    from subprocess import Popen, PIPE, STDOUT
    p = Popen(cmd, shell=True, stdout=PIPE, stderr=STDOUT)
    output, dummy = p.communicate()
    failure = p.returncode
    if failure:
        print 'Command failed:', cmd; sys.exit(1)






Code for interpreting the grabbed output 

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errors = {'dt': dt_values, 1: [], 0: [], 0.5: []}
for line in output.splitlines():
    words = line.split()
    if words[0] in ('0.0', '0.5', '1.0'):  # line with E?
        # typical line: 0.0   1.25:    7.463E+00
        theta = float(words[0])
        E = float(words[2])
        errors[theta].append(E)

Next: plot \( E \) versus \( \Delta t \) for \( \theta=0,0.5,1 \)

Complete program: experiments/decay_exper1.py. Fine recipe for






Making a report






Publishing a complete project