Technische Universität München
Lehrstuhl für Informatik V
Dr. Tobias Neckel
Ivana Buha, Iryna Burak
SS 2024
April 17, 2024
Algorithms for Uncertainty
Quantification
Tutorial 1: Python overview + ODE numerics
In this worksheet, we do a short overview of programming in python. Note: Everything
that follows is a lot easier if you use a Unix-based operating system on your machine. If
you are using Windows then we recommend you to use Anaconda. We recommend you
to create a virtual environment for this course.
We assume that you are familiar with the basics concepts of computer programming:
• data types
• variables
• loops
• arrays
• functions
• etc.
If you are new to programming and you want to cover the basic concepts, see e.g.
tutorialspoint.com/computer programming/computer programming basics.htm.
Short overview of programming with python
Throughout these tutorials, we use python 3.x. How to use python1 ?
• directly in the command line: simply type python
1
extensive python 3.x documentation at https://docs.python.org/3/
1
• via Jupyter2 notebooks
• via an IDE, e.g. Spyder, Eclipse + PyDev, PyCharm, VisualStudio
python is
• strongly typed: types are enforced
• dynamically, implicitly typed: you don’t need to declare variables
• case sensitive: a and A are different variables
• object-oriented: everything is an object
Data structures
Build-in data structures
• lists: similar to 1D arrays
• dictionaries: hash-tables (key-value pairs)
• tuples: immutable 1D arrays
• sets: since python 2.5
>>> # l i s t example
>>> m y l i s t = [ ’ a ’ , 1 , 2 , [ 1 , 2 ] ]
>>> m y l i s t [ 1 ] = 22
>>> m y l i s t
[ 1 , 22 , ’ a ’ , [ 1 , 2 ] ]
>>> # t u p l e example
>>> mytuple = ( 1 , 2 )
>>> mytuple [ 1 ] = 2
Traceback ( most r e c e n t c a l l l a s t ) :
F i l e ”<s t d i n >” , l i n e 1 , in <module>
TypeError : ’ t u p l e ’ object does not s u p p o r t item a s s i g n m e n t
>>> # d i c t i o n a r y example
>>> mydict = { 1 : ’ a ’ , 0 : ’ b ’ }
>>> mydict [ 0 ]
2
http://jupyter.org/
2
’b ’
>>> mydict . keys ( )
[0 , 1]
>>> mydict . v a l u e s ( )
[ ’b ’ , ’a ’ ]
>>> mydict [ 5 ] = ’ f o o ’
>>> mydict
{0: ’a ’ , 1: ’b ’ , 5: ’ foo ’}
>>> # s e t example
>>> m y l i s t = [ 1 , 2 , 2 , 3 ]
>>> set ( m y l i s t )
set ( [ 1 , 2 , 3 ] )
>>> # l i s t s l i c i n g
>>> m y l i s t = [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 ]
>>> m y l i s t [ 0 : − 1 ]
[1 , 2 , 3 , 4 , 5 , 6 , 7 , 8]
>>> m y l i s t [ 2 : 4 ]
[3 , 4]
>>> m y l i s t [ −1: −4: −1]
[9 , 8 , 7]
>>> m y l i s t [ : 5 ]
[1 , 2 , 3 , 4 , 5]
Flow control statements
Flow control statements in python: if, for, and while. To obtain a list of numbers,
use range<number>
>>> for i in range ( 5 ) :
...
if i % 2:
...
print ( i )
...
else :
...
print ( i )
...
−1
1
1
3
3
3
Functions
Functions are declared with the def keyword. Optional arguments are set in the function
declaration. In python, functions can return a tuple (this way, you can effectively return
multiple values)
>>> def i n c r e a s e b y o n e ( x ) :
...
return x + 1
...
>>> i n c r e a s e b y o n e ( 2 )
3
>>> i n c r e a s e b y o n e ( 2 . 3 )
3.3
>>> n e w i n c r e a s e b y o n e = lambda x : ( x + 1 , 0 )
>>> r e s = n e w i n c r e a s e b y o n e ( 1 )
>>> r e s
(2 , 0)
>>> r e s [ 1 ] = 2
Traceback ( most r e c e n t c a l l l a s t ) :
F i l e ”<s t d i n >” , l i n e 1 , in <module>
TypeError : ’ t u p l e ’ object does not s u p p o r t item a s s i g n m e n t
Use of external libraries
>>> # import e n t i r e l i b r a r y
>>> import [ libname ]
>>> # import s p e c i f i c f u n c t i o n s / p a c k a g e s
>>> from [ libname ] import [ funcname | packagename ]
Object-oriented programming
Classes are declared with the class keyword. Private variables and methods can be
declared, however, by convention, this is not enforced by the language.
>>> c l a s s MyClass :
init ( self ) :
...
def
...
print ( ’ t h i s i s an i n s t a n c e o f MyClass ’ )
4
...
s e l f . v a r i a b l e = 10
...
def myfunction ( s e l f , x ) :
...
return 2∗ x
...
>>> i n s t a n c e = MyClass ( )
t h i s i s an i n s t a n c e o f MyClass
>>> i n s t a n c e . myfunction ( 1 0 )
20
>>> i n s t a n c e . v a r i a b l e
10
>>> # i n h e r i t a n c e example
>>> c l a s s SubClass ( MyClass ) :
init ( self ) :
...
def
...
super ( ) . i n i t ( )
...
print ( ’ t h i s i s an i n s t a n c e o f t he s u b c l a s s ’ )
...
>>> i n s t a n c e=SubClass ( )
t h i s i s an i n s t a n c e o f MyClass
t h i s i s an i n s t a n c e o f th e s u b c l a s s
Polymorphism is also possible
>>> c l a s s Animal :
i n i t ( s e l f , name ) :
...
def
...
s e l f . name = name
...
def t a l k ( s e l f ) :
...
r a i s e NotImplementedError ( ” A b s t r a c t method must
be implemented ” )
...
>>> c l a s s Cat ( Animal ) :
...
def t a l k ( s e l f ) :
...
return ’Meow ! ’
...
>>> c l a s s Dog ( Animal ) :
...
def t a l k ( s e l f ) :
...
return ’ Woof ! Woof ! ’
...
>>> c l a s s Human( Animal ) :
...
def bark ( s e l f ) :
5
...
print ( ’UQ i s t he b e s t ’ )
...
>>> a n i m a l s = [ Cat ( ’Mr . P u r r i f y ’ ) , Dog ( ’ L a s s i e ’ ) , Human( ’me ’ ) ]
>>> for animal in a n i m a l s :
...
print ( animal . name + ’ : ’ + animal . t a l k ( ) )
...
Mr . P u r r i f y : Meow !
L a s s i e : Woof ! Woof !
Traceback ( most r e c e n t c a l l l a s t ) :
F i l e ”<s t d i n >” , l i n e 2 , in <module>
F i l e ”<s t d i n >” , l i n e 5 , in t a l k
NotImplementedError : A b s t r a c t method must be implemented
Scientific computing with python
Since python is an interpreted and dynamically typed programming language, python
programs can be slow compared to compiled statically typed programming languages,
such as C and Fortran.
However, using suitable libraries, python can be used for scientific computing3 .
Numpy - multidimensional data arrays
package that provides high-performance vector, matrix and higher-dimensional data
structures for python.
>>> import numpy as np
>>> # do c l e v e r t h i n g s
SciPy - Library of scientific algorithms
It includes
• Special functions (scipy.special)
• Integration (scipy.integrate)
• Optimization (scipy.optimize)
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some materials on scientific computing https://github.com/jrjohansson/scientific-python-lectures
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• Interpolation (scipy.interpolate)
• Fourier Transforms (scipy.fftpack)
• Signal Processing (scipy.signal)
• Linear Algebra (scipy.linalg)
• Sparse Eigenvalue Problems (scipy.sparse)
• Statistics (scipy.stats)
• Multi-dimensional image processing (scipy.ndimage)
• File IO (scipy.io)
matplotlib - 2D and 3D plotting
Using Fortran and C code
In python it is relatively easy to call out to libraries with compiled C or Fortran code
• Fortran: F2PY
• C: ctypes, Cython
Tools for high-performance computing applications
• thread parallelism: the multiprocessing package
• IPython parallel
• MPI: mpi4py
• opencl: pyopencl
• etc.
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Assignment
A linear damped oscillator is a modeled by a second order ODE
 2
d y
dy

 dt2 (t) + c dt (t) + ky(t) = f cos(ωt)
y(0) = y0

 dy
(0) = y1 ,
dt
(1)
where c is the damping coefficient, k the spring constant, f the forcing amplitude, ω the
frequency, y0 represents the initial position, whereas y1 is the initial velocity.
Assignment 1
Transform the above equation in a first order system of ODEs. Hint: Use helper functions
z0 (t) := y(t),
dy(t)
,
z1 (t) :=
dt
..
.
zn−1 (t) :=
dy n−1 (t)
dtn−1
and transform into system
dz0
:= z1 ,
dt
dz1
:= z2 ,
dt
..
.
f (t, z0 ,
dz0 dz1
dzn−1
,
,··· ,
) := 0.
dt dt
dt
Solution:
z0 (t) := y(t),
dy(t)
dz0 (t)
z1 (t) :=
=
,
dt
dt
d2 y(t)
dz1 (t)
z2 (t) :=
=
.
2
dt
dt
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Write down system:
dz0 (t)
= z1 (t)
dt
dz1 (t)
ż1 :=
= −cz1 (t) − kz0 (t) + f cos(ωt)
dt
ż0 :=
or in matrix/vector notation:
ż0
0
1
z0 (t)
0
=
+
ż1
−k −c
z1 (t)
f cos(ωt)
Assignment 2
Considering t ∈ [0, 20], c = 0.5, k = 2.0, f = 0.5, ω = 1.05, y0 = 0.5, y1 = 0.0 and
the setup as in Assignment 1, discretize the oscillator using explicit Euler time on the
system with ∆t = 0.1, 0.05, 0.01. You can use standard python lists or numpy lists.
Change ∆t = 0.5, what do you observe? Download ex2 framework.py from Moodle
and implement the unimplemented part. Solution: See code Code Solution.py
Assignment 3
With the same setup, solve Equation 1 using the odeint4 function from scipy.integrate.
Download ex3 framework.py from Moodle and implement the unimplemented part.
Solution: See code Code Solution.py
4
see https://docs.scipy.org/doc/scipy-0.17.0/reference/generated/scipy.integrate.odeint.html
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