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What is Scientific . . .
Why Numerical . . .
From Phenomena to . . .
Mathematical Modelling
Levels of Point of View
Numerical Treatment . . .
Related issues (What . . .
Literature
Page 1 of 13
Introduction to Scientific Computing
1. DefinitionMichael Bader
Introduction to Scientific Computing
What is Scientific Computing?
October 22, 2003
What is Scientific . . .
Why Numerical . . .
From Phenomena to . . .
Mathematical Modelling
Levels of Point of View
Numerical Treatment . . .
Related issues (What . . .
Literature
Page 2 of 13
Introduction to Scientific Computing
1. DefinitionMichael Bader
1. What is Scientific Computing?
• mathematical and informatical basis of numerical simulation
• reconstruction or prediction of phenomena and processes, esp.from science and engineering, on supercomputers
• third way to obtain knowledge apart from theory and experiment
?
Exp
erim
ent
Sim
ula
tio
n
Th
eory
• transdisciplinary: mathematics + informatics + field of applica-tion!
• Objectives depend on actual task of simulation:
– reconstructand understandknown scenarios (natural disas-ters)
– optimizeknown scenarios (technical processes)
– predictunknown scenarios (weather, new materials)
What is Scientific . . .
Why Numerical . . .
From Phenomena to . . .
Mathematical Modelling
Levels of Point of View
Numerical Treatment . . .
Related issues (What . . .
Literature
Page 3 of 13
Introduction to Scientific Computing
1. DefinitionMichael Bader
2. Why Numerical Simulations?
2.1. since experiments are sometimes impossible:
• astrophysics: life cycle of galaxies etc.
• climate research: Gulf Stream, greenhouse effect etc.
What is Scientific . . .
Why Numerical . . .
From Phenomena to . . .
Mathematical Modelling
Levels of Point of View
Numerical Treatment . . .
Related issues (What . . .
Literature
Page 4 of 13
Introduction to Scientific Computing
1. DefinitionMichael Bader
• weather forecast: tornadoes – where, when, and how strong?
• simulation of stockmarkets, or predicting economic effects (in-troducing the “Euro”)
What is Scientific . . .
Why Numerical . . .
From Phenomena to . . .
Mathematical Modelling
Levels of Point of View
Numerical Treatment . . .
Related issues (What . . .
Literature
Page 5 of 13
Introduction to Scientific Computing
1. DefinitionMichael Bader
2.2. since experiments are sometimes very unwelcome
• security of nuclear power plants, tests of nuclear weapons
• statics: stability of buildings etc.
• propagation of harmful substances
What is Scientific . . .
Why Numerical . . .
From Phenomena to . . .
Mathematical Modelling
Levels of Point of View
Numerical Treatment . . .
Related issues (What . . .
Literature
Page 6 of 13
Introduction to Scientific Computing
1. DefinitionMichael Bader
2.3. since experiments can be extremely costly
• aerodynamics, turbulence: objects in a wind tunnel and so on
• car industry: crash tests
What is Scientific . . .
Why Numerical . . .
From Phenomena to . . .
Mathematical Modelling
Levels of Point of View
Numerical Treatment . . .
Related issues (What . . .
Literature
Page 7 of 13
Introduction to Scientific Computing
1. DefinitionMichael Bader
• analysis and study of proteins
• Molecular dynamics: crystal structure, macromolecules
What is Scientific . . .
Why Numerical . . .
From Phenomena to . . .
Mathematical Modelling
Levels of Point of View
Numerical Treatment . . .
Related issues (What . . .
Literature
Page 8 of 13
Introduction to Scientific Computing
1. DefinitionMichael Bader
3. From Phenomena to Predictions
phenomenon, process etc.
mathematical model?
modelling
numerical algorithm?
numerical treatment
simulation code?
implementation
results to interpret?
visualization
�����
HHHHj embedding
statement tool
-
-
-validation
Makes up the roadmap for our lecture!
What is Scientific . . .
Why Numerical . . .
From Phenomena to . . .
Mathematical Modelling
Levels of Point of View
Numerical Treatment . . .
Related issues (What . . .
Literature
Page 9 of 13
Introduction to Scientific Computing
1. DefinitionMichael Bader
4. Mathematical Modelling
• model as a (simplifying) formal abstraction of reality
• issues when deriving a mathematical model:
– Which quantities are of influence; how important are they?
– What relations exist between them? "Which type of mathe-matics?"
– What is the given task (solve, optimize, etc.)?
• issues when analyzing a mathematical model:
– Does a solution exist? Ist it unique?
– Do the results depend continuously on the input data?
– How accurate is the model, what can be represented?
– Is the model well-suited for a numerical treatment?
• There is not one correct model, but several possible!
• model hierarchy: accuracy vs. complexity
Our focus: Ordinary/Partial Differential Equations
What is Scientific . . .
Why Numerical . . .
From Phenomena to . . .
Mathematical Modelling
Levels of Point of View
Numerical Treatment . . .
Related issues (What . . .
Literature
Page 10 of 13
Introduction to Scientific Computing
1. DefinitionMichael Bader
5. Simulating Men: Levels of Point of View
issue level of resolution model basis (e.g.!)global increasein population
countries, regions population dynamics
local increase inpopulation
villages, individuals population dynamics
man circulations, organs system simulatorblood circulation pump/channels/valves network simulatorheart blood cells continuumcell macro molecules continuummacromolecules
atoms molecular dynamics
atoms electrons or finer quantum mechanics
What is Scientific . . .
Why Numerical . . .
From Phenomena to . . .
Mathematical Modelling
Levels of Point of View
Numerical Treatment . . .
Related issues (What . . .
Literature
Page 11 of 13
Introduction to Scientific Computing
1. DefinitionMichael Bader
6. Numerical Treatment of Models
• In practice, models can typically not be solved analytically
• numerical (approximate and computer-based) methods!
• the numerical part is non-trivial:
– find appropriate representation on the computer (arrays in-stead of functions)
– find appropriate solution method
Topics in this lecture:
• Numerical methods for ordinary differential equations (ODE)
• Numerical methods for partial differential equations (PDE)
What is Scientific . . .
Why Numerical . . .
From Phenomena to . . .
Mathematical Modelling
Levels of Point of View
Numerical Treatment . . .
Related issues (What . . .
Literature
Page 12 of 13
Introduction to Scientific Computing
1. DefinitionMichael Bader
7. Related issues (What else has to be done?)
• a numerical algorithm is not yet an efficient code
• the implementation is crucial:
– platform microprocessor: pipelining, cache memory
– platform supercomputer: vector/parallel/vector-parallel/cluster,distributed/shared/virtually shared memory
– suitable data structures and organization principles (hierar-chy, recurrences)
– potential of (automated) parallelization, communication
– today: software engineering important in simulation con-text, too
• “MATLAB-numerics” is not sufficient for doing relevant numeri-cal simulations!
• we need “plug-and-play tools”: embedding
• interpretation of tons of data requires visualization
What is Scientific . . .
Why Numerical . . .
From Phenomena to . . .
Mathematical Modelling
Levels of Point of View
Numerical Treatment . . .
Related issues (What . . .
Literature
Page 13 of 13
Introduction to Scientific Computing
1. DefinitionMichael Bader
References
[1] Walter Gander and Jiri Hrebícek. Solving Problems in ScientificComputing Using Maple and MATLAB. Springer-Verlag, Berlin,Germany / Heidelberg, Germany / London, UK / etc., secondedition, 1995.
[2] Werner Krabs. Mathematische Modellierung. Eine Einführung indie Problematik. Teubner-Verlag, 1997.
[3] Gene H. Golub and James M. Ortega. Scientific Computing andDifferential Equations. Academic Press, Boston, MA, USA, 1992.
[4] Jack J. Dongarra, Iain S. Duff, Danny C. Sorensen, and Henk A.van der Vorst. Numerical linear algebra for high-performancecomputers. Society for Industrial and Applied Mathematics(SIAM), Philadelphia, PA, USA, 1998.
[5] Kai Hwang. Advanced Computer Architecture: Parallelism, Scal-ability, Programmability. McGraw-Hill, New York, 1993.
[6] Michael Griebel, Thomas Dornseifer, and Tilman Neunhoeffer.Numerical Simulation in Fluid Dynamics: A Practical Introduc-tion. Society for Industrial and Applied Mathematics (SIAM),Philadelphia, PA, USA, 1997.