Introduction to Computational Science Summary CHoogteijling
Introduction to Computational Science
Contents
1 Modelling and Simulation 2
2 Cellular Automata 2
3 Network Models 9
1
,Introduction to Computational Science Summary CHoogteijling
1 Modelling and Simulation
The 3 pillars of science are:
1. Theory
2. Experimentation
3. Modelling and simulation
The limitations of a model:
• The model has a domain of validity.
• The model only represents a part of the original system.
• The models output will not exactly match that of the original system, it has limited accuracy.
We can use models because the physical system is not available, the experiment may be dangerous,
it is too expensive to experiment, the time scale of the system is too short or long, the control
variables may be inaccessible, or to suppress disturbances.
Definitions in modelling and simulation:
• Validation. Checking whether the model reproduces the real data.
• Verification. Checking whether the simulation is correct, whether the simulation program
does what it is supposed to do.
• System. A potential source of data.
• Experiment. The process of extracting data from a system by exerting it through its inputs.
• Model. A model for a system and an experiment is anything to which an experiment can be
applied in order to answer questions about the system.
• Simulation. An experiment performed on a model.
• Experimental frame. A specification of the conditions under which the system is observed is
experimented with.
2 Cellular Automata
2.1 Types of models
In continuous-time models the state changes continuously over time. Within a finite time span,
the state variables change their values infinitely often. It can be represented by a set of differential
equations.
In discrete-time models the time-axis is discretized. It is represented by finite difference equa-
tions: uk+1 = u(tk+1 = f (uk , tk ).
2
, Introduction to Computational Science Summary CHoogteijling
In discrete-event models events can happen at any time and state changes by jumps in time.
It is modelled through probabilistic functions. The state variables change at discrete events. Both
time axis and state axis are continuous. In a finite time span only a finite number of changes may
occur.
Examples of real-life applications of continuous, discrete-time, and discrete-event models are:
• Continuous models can be used to analyse the dynamics of financial markets, such as stock
prices or interest rates.
• Discrete-time models can be used to predict future values of financial variables such as stock
prices or interest rates using time-series models.
• Discrete-event models can be used to simulate financial markets, investment strategies, and
risk analysis.
!
'
i ! ! ! 1
~ \ 1 't 1 r- '! '
1 i
! i
'' ~
NA r
.,.
·7 n-7--r - ~ "
'i {
r
9.,- - -
i
. . ll ~s h '-'·
1
1
. -- ·- - - -- - - -
~
--
1 !'
! : - ·- ----·· --+--_.,._-+_
i , .,~ :t1 \ 41 1'IV: \ ?"'V\: 1
l .
, 1 , , i l 1 l 1 1
l --1'90~ 'Pr v , . j - 1
: i i \ 1: i
, d'W~
1
\
1 1 1 l 1 1 l 1 1 1 i , \
Figure 1: Time-state plots of continuous, discrete-time, and discrete-event models.
2.2 The cellular automaton
A cellular automaton is a N -dimensional grid of cells and it has discrete states (finite number).
The new state of a cell depends on, its own state, states of neighbour cells, and rules that are
applied to all cells simultaneously, except in asynchronous CA.
We study cellular automata, for the emergence of complex, systemic behaviour out of simple local
dynamics, as original models of fundamental physics, discrete dynamical system simulators, and as
powerful computation engines.
r 1 range
N 3 neighbourhood
k 2 ∈ 0, 1 amount of states
L ∈ 000, 001, . . . alphabet
|L| = m kN = 8 size of alphabet
N
kk amount of rules
Table 1: The characteristics of a CA.
3
Introduction to Computational Science
Contents
1 Modelling and Simulation 2
2 Cellular Automata 2
3 Network Models 9
1
,Introduction to Computational Science Summary CHoogteijling
1 Modelling and Simulation
The 3 pillars of science are:
1. Theory
2. Experimentation
3. Modelling and simulation
The limitations of a model:
• The model has a domain of validity.
• The model only represents a part of the original system.
• The models output will not exactly match that of the original system, it has limited accuracy.
We can use models because the physical system is not available, the experiment may be dangerous,
it is too expensive to experiment, the time scale of the system is too short or long, the control
variables may be inaccessible, or to suppress disturbances.
Definitions in modelling and simulation:
• Validation. Checking whether the model reproduces the real data.
• Verification. Checking whether the simulation is correct, whether the simulation program
does what it is supposed to do.
• System. A potential source of data.
• Experiment. The process of extracting data from a system by exerting it through its inputs.
• Model. A model for a system and an experiment is anything to which an experiment can be
applied in order to answer questions about the system.
• Simulation. An experiment performed on a model.
• Experimental frame. A specification of the conditions under which the system is observed is
experimented with.
2 Cellular Automata
2.1 Types of models
In continuous-time models the state changes continuously over time. Within a finite time span,
the state variables change their values infinitely often. It can be represented by a set of differential
equations.
In discrete-time models the time-axis is discretized. It is represented by finite difference equa-
tions: uk+1 = u(tk+1 = f (uk , tk ).
2
, Introduction to Computational Science Summary CHoogteijling
In discrete-event models events can happen at any time and state changes by jumps in time.
It is modelled through probabilistic functions. The state variables change at discrete events. Both
time axis and state axis are continuous. In a finite time span only a finite number of changes may
occur.
Examples of real-life applications of continuous, discrete-time, and discrete-event models are:
• Continuous models can be used to analyse the dynamics of financial markets, such as stock
prices or interest rates.
• Discrete-time models can be used to predict future values of financial variables such as stock
prices or interest rates using time-series models.
• Discrete-event models can be used to simulate financial markets, investment strategies, and
risk analysis.
!
'
i ! ! ! 1
~ \ 1 't 1 r- '! '
1 i
! i
'' ~
NA r
.,.
·7 n-7--r - ~ "
'i {
r
9.,- - -
i
. . ll ~s h '-'·
1
1
. -- ·- - - -- - - -
~
--
1 !'
! : - ·- ----·· --+--_.,._-+_
i , .,~ :t1 \ 41 1'IV: \ ?"'V\: 1
l .
, 1 , , i l 1 l 1 1
l --1'90~ 'Pr v , . j - 1
: i i \ 1: i
, d'W~
1
\
1 1 1 l 1 1 l 1 1 1 i , \
Figure 1: Time-state plots of continuous, discrete-time, and discrete-event models.
2.2 The cellular automaton
A cellular automaton is a N -dimensional grid of cells and it has discrete states (finite number).
The new state of a cell depends on, its own state, states of neighbour cells, and rules that are
applied to all cells simultaneously, except in asynchronous CA.
We study cellular automata, for the emergence of complex, systemic behaviour out of simple local
dynamics, as original models of fundamental physics, discrete dynamical system simulators, and as
powerful computation engines.
r 1 range
N 3 neighbourhood
k 2 ∈ 0, 1 amount of states
L ∈ 000, 001, . . . alphabet
|L| = m kN = 8 size of alphabet
N
kk amount of rules
Table 1: The characteristics of a CA.
3
