1. Monday, Week 1: Methodology
STE 6239 Simulering
Monday, Week 1:
Methodology

2. 1.Introduction: what are modelling and simulation, and why do we need them?
1.1. Intuitive idea about modelling and simulation
Our life motivation: our goals in life (depend on ’parameters’ such as: moral, intellectual, material, ethical values, national specifics, social standing, cultural background, etc.)
Decision-making: very important for achieving our goals (based on selection of steps towards the goal, which is achieved by modelling and simulation of realistic situations)
Modelling and simulation ability is a key measure for the level of intellect. A less intelligent person constructs and simulates mentally simpler models, and relies more on the trial and error approach. A more intelligent person is able to process more advanced models, makes fewer errors thanks to better prediction, and ’moves by larger and faster steps’ to achieving higher goals.
Our daily survival depends on our skills for modelling and simulation because we need to avoid various dangers, which is achieved by prediction. A few daily examples:

3. (continued)
All inventions, as steps in the continuous process of technological progress, are the product of more or less advanced modelling and simulation of a future situation. Examples:
Our intuitive models are more or less abstract constructions reflecting aspects of the material universe, with abstract data input, while their simulation is the functioning in time of concrete implementations of the models, with concrete data inputs of the respective abstract type. Examples:

4. (continued)
By acquiring knowledge and experience, we can improve our ability for modelling and simulation (valid for the single individual and for the society as a whole). The history of technological progress is also the history of improving human mastery in modelling and simulation, characterized by increasing fidelity (closeness to the real phenomenon) of the model and, as a consequence, its increasing algorithmic and computational complexity. The biggest jumps in the methodology and level of detail of modelling and simulations are usually correlated with the appearance of new, qualitatively more powerful, computational devices, e. g., in the recent decades, this is usually the appearance of a next generation of computer chips (Details about Moore’s Law and other computer technology related upward development curves).
Scientific models in the natural and engineering sciences and industry are characterized by the highest level of fidelity, thanks to the methodology of mathematical modelling and numerical simulation, which are the common and uniting links between all ’exact’ branches of science.

5. The purpose of the present course:
To teach you the basics of numerical simulation of mathematical models as a unifying fundamental methodology common for all natural and engineering sciences and most of their industrial applications.

6. Important types of systems
1.2. Formalization of the concepts of a mathematical model and its simulation
The stepping stones of the models produced by human thinking are the concepts. Most of the concepts are being defined via other, simpler concepts. Since our thinking is finite (because such are our lives), there must be some simplest concepts which are only based on our intuitive understanding of them. These concepts are called axiomatic.
(Note: Some statements involving axiomatic objects are also called axiomatic, or simply axioms.
Definition: A set, together with some rules describing relations between its subsets is called a system.
(Note that in this setting the concept of a ’system’ is NOT axiomatic, but the concept of a ’rule describing relation’ IS! In fact, the concept of a relation is often described in terms of even simpler concepts, but we do not need these details in this course. The concept of relation is sufficiently intuitive in itself!)
Important types of systems

7. (Continued)
General definition: The concept of a mathematical model of a given system S most generally refers to a system M such that:
The rules and relations between the subsets of M can be explored by available mathematical methods
M approximates S according to some criteria ensuring:
Approximation of the subsets of S by subsets of M
Approximation of the rules and relations for the system S by the rules and relations for the system M
General definition: The concept of Simulation of the system S by the model M refers most generally to:
Any procedure which:
involves the rules and relations for the system M
contains a verification sub-procedure which evaluates the quality of approximation of S by M, i.e., the quality of the mathematical model
The results of any such procedure
Remark: Mathematical modelling and simulation is an inherently iterative process: the results of the simulation of a given model can be used to improve this model, and so on

8. 1.3. Example: Mathematical models of dynamical systems
In the course we shall see examples of both continuous-time and discrete-time dynamical systems. For simplicity, here we shall consider only the case of continuous time.
Stage 1: collecting empirical (observational, experimental, etc.) data about the target dynamical system (physical or conceptual).
Stage 2: based on the data from Stage 1, design of the mathematical model (a conceptual system), consisting generally of:
1. The most important variables determining the state (at every fixed moment) and behaviour (the evolution of the state with time) of the dynamical system.
2. The quantitative relations (in mathematical form) governing the variations of the variables identified in the previous item 1.

9. (continued)
More concretely, the construction of the mathematical model goes through two main stages:
A. Identifying the parameters of the state. In general, they are functions of the independent variables of the system:
where t is time, the x’s are the idependent variables, and the q’s are the parameters of the state (or dependent parameters) of the system. For the description of physical processes, n=3, and the independent variables are most often the time and the three spatial coordinates x,y,z. The q’s are computed in the form (1) by solving the equations (2) of the mathematical model:

10. (continued)
which are m equations for n+m variables, solved with respect to the m q-variables.
B. Identifying the parameters of the system, i.e., additional parameters of the system which characterize the properties of the system (e.g., the mass of parts of the system) and the influence of the external environment, if the system is open, e.g., the external forces acting on the system.
Example: a system of several bodies moving in their own gravitational field. The masses of the bodies are parameters of the system, which will be have differently when the masses are different. The external forces will be 0
(because the system is closed).

11. 1.4. General definition of a mathematical model of a dynamical system
Definition: The mathematical model of a dynamical system is an equation (or a system of equations), which connects the parameters of the state and their derivatives in the independent variables (time and/or other variables).
Typically, the derivation of this mathematical model is based on physical laws or other rules described in terms of the parameters of the system.
Remark: In the case of discrete-event dynamical systems, ’derivatives’ in the above definition should be replaced by ’finite differences’ or ’divided differences’ depending on whether the step in time is uniform or not
Example: movement of a material point with mass m, under the external force
F(t)=(f(t),g(t),h(t)).

12. (Example, continued)
Step A. Determining the parameters of the state. The position of the point is r(t)=(x(t),y(t),z(t)) and it depends only on the time t. The parameters of the state in this case are x,y,z, while t is the only independent variable.
Step B. Determining the parameters of the system. In this case they are the point mass m (characterizing the internal properties of the system) and the external force F applied to the point. In general, F= F(t,r,v) (e.g., for a string), where v = dr /dt =(U,V,W) is the velocity of the point.
Step C. Derivation of the mathematical model:

13. (Example, continued) Assuming that the point mass m is constant during the process,
which is equivalent to
with the following initial conditions
which is a typical Cauchy initial value problem. We can now investigate by mathematical methods the conditions for existance and uniqueness of the solution of this system, look for periodical solutions, and so on. Note that the solution, when it exists and is unique, is of the form x=x(t),y=y(t),z=z(t), which is of the prescribed form for m=3,n=1.

14. 1.5. General block scheme of the joint process of pre- and post-processing, mathematical modelling and simulation of a dynamical system
Modelling/Simulation
Phenomenolgical, observational,
empirical and experimental
study of the target system
Begin
Eventual
modification
of empirical and
experimental
setting
Eventual
modification of
experimental
setting
Definition domains
of the independent
variables
Selection of
parameters of state
Definition of the
parameters of
the system
Formulation
of the model
Eventually
modified/new
mathematical model
and analytical method
Eventually
modified/new
mathematical model
and numerical
simulation method
Analysis of
the model N
Analyt.
method Y
OK? N
Numer.
Y method
OK?
’Pure mathematics’:
Exact or approximate
solution of the model
in abstract analytic form
’Applied mathematics’:
numerical or analogue
approximate solution in
concrete context
Computer
-aided
symbolic
computation
(CPUs,
GPUs)
’Pure mathematics’:
analytic investigation
of solution, very high
quality, in general context,
but narrow range of
applicability
Computer-aided
advanced methods
for visualization
with approximation
(GPUs, CPUs)
’Applied mathematics’:
numerical simulation
in concrete context,
variable quality, very
broad range of
applicability
Computer
number-
crunching
power
(CPUs,
GPUs)
Analytic verification,
general context.
narrow range
of applicability
Practical approximate
verification, concrete
context, very broad
range of applicability
Generalizations,
extensions,
enhancements
of the model
Tuning the
parameters
of the simulation
or modified/new
numerical method
Modified or new
analytic method N Y
Y N
N End? Y
End

15. 1.5.1. Some comments to the Block Scheme in Item 1.5:
The red part corresponds to the Modelling stage
The yellow and green parts correspond to the Simulation stage
The yellow part corresponds to theoretical simulation using analytic methods, symbolic computation of closed-form solutions, etc.
The green part corresponds to numerical simulation using approximation and computational power
The simulation stage
includes a verification substage
Verification is of key importance for the iterative process of improving the model
Important part of the verification substage is scientific visualization (more than 90% of the information acquired by the senses is visual, and this type of information is easiest to calibrate and standardize)

16. 1.6. Classification of mathematical models
Dynamic vs. Static: dependence/independence of a time variable
Example: a model describing the geometric form of a curve is static
Example: a model describing a curve as a trajectory of a moving object is dynamic (geometric form + parametrization)
Linear vs. Nonlinear: linearity/nonlinearity of the mathematical equations involved
Deterministic vs. Indeterministic (Stochastic)
Deterministic: same behaviour for same input data
Indeterministic (stochastic): different (random) behaviour for same input data
Probabilistic: well-posed (correctly posed) direct (non-inverse) stochastic problems
Statistical: ill-posed (incorrectly posed) inverse stochastic problems
White-box vs. Gray-box vs. Black-box models: depending on the amount of information about the model
White-box models: complete information is available how input data are transformed in the model into output results
Example: math. equations of dynamical systems from Items 1.2-5
Gray-box models: only incomplete partial information is available on how input data are transformed in the model into output results
Example: black-and-white images are replaced by grayscale ones;
Example: sets with exact boundaries are replaced by fuzzy sets, etc.
Example: uncertainty principles
Black-box models: no information is available at all how input data are transformed in the model into output results; this means that all information about this problem is based only on the input data and output results
Example: models based on training of neural networks
Example: statistical models

17. 1.6.1. Classification of variables in ’Box’ models
A priori information
Input variables
In stochastic models:
Cumulative distribution
Density of distribution
Probability
Expectation of random variables
Variance of random variables
etc.
A posteriori information
Output variables
Conditional cumulative distribution
Conditional density
Conditional probability
Conditional expectation
Hidden variables:
appear when a Black-box model is considered as a White-box model.
The internal structure and functionality of the White-box model is described using additional parameters – variables of the White box model which are unavailable for the Black-box model
These additional parameters (called hidden variables of the Black-box model) have to be determined and estimated based only on the information available for the Black-box model (the available a priori and a posteriori information)
This estimation is typically a very difficult ill-posed inverse problem
Example: hidden variables in quantum mechanics

18. 1.7. Classification of computer-aided simulations
Simulations on analogue (analog) computers
An analogue computer – a physical device simulating the target prosess (the phenomenon of interest) using electronic, mechanical or other physical phenomena described by the same mathematical model, but cheaper and simpler to implement.
Historically appears before the electronic digital computer. Mechanical versions are known since antiquity
Examples:
the Antikythera mechanism
Kircchoff electrical chains modelling the flow in tubes and underground reservoirs (e.g., applied to petroleum research and other physiacl phenomena governed by systems of differential equations)
Astrolabes and giroscopes
A hydraulic model of the economy of the United Kingdom
The wind tunnel
The slide rule
Etc.
Limitations of analogue computers
The range of parameters in the analogous process used is often much narrower than the range of the parameters in the target process of interest
Noises, disturbances and small perturbations in the analogous process which are unspecific for the target process
The possibility to upgrade, extend, generalize and enhance the underlying mathematical model is very limited, or inexistant

19. (continued) 1.7.2. Simulations on electronic digital computers
Convention: in this course, NUMERICAL SIMULATION is used as synonym of SIMULATION ON AN ELECTRONIC DIGITAL COMPUTER
Criteria for classification of simulations
(A) Type of the mathematical model (dynamical systems, black box, white box, etc.)
(B) Fidelity of the mathematical model (how well the model describes the target phenomenon)
(C) Fidelity of the approximate model used in the simulations (how good the numerical approximation is to the mathematical model, independent of the fidelity of the latter, described in (B))
(D) Algorithmic complexity of the mathematical model (the typical price to pay for high (B) is high (D))
(E) Computational complexity of the approximate model used in the simulations (the typical price to pay for high (B) and/or high (C) is high (E))
(F) Quality of the verification sub-procedure (scientific visualization; interactive verification; automatic verification)
(G) Type of computing architecture (sequential, parallel; CPU-based, GPU-based, etc.)

20. (Continued)
Examples
Continual or discrete-event simulation - criterion (A): dynamical systems
Stochastic or deterministic (and, as a special case of deterministic – chaotic) simulation – criterion (A): black-box and white-box math.models
Modelling fluid motion, convection, combustion, etc. by Navier-Stokes and similar PDEs: high (B) leading to high (D); to keep the final result good (C) and (E) must also be high
Flight simulators – ’good’ (high (B) and (C)) but ’expensive’ (high (D) and (E))
Special effects in computer games – typically, low fidelity (’bad’) (low (B) and/or (C)) but ’cheap’ (low (D) and (E))
Verification – criterion (F):
Interactive verification: by human decision-taking
Automatic verification: cross validation = any procedure of separating the data set into several disjoint subsets, running the simulation on each of the subsets and comparing the results obtained for the different subsets
Local or distributed simulation – criterion (G): typically, sequential is local, parallel is distributed
Agent-based simulation (Particle simulation): the global mathematical model consists of local sub-models which interact together. (For simplicity, here we assume that the mathematical model is the same as the approximated model used in the simulations.) Typically:
The local sub-models (hence, also the global model) have low (D)
The local sub-models have low (E)
The global model has high (E) (because there are MANY local sub-models)