introduction to scientific computing · what is scientiﬁc... why numerical... from phenomena...

What is Scientific . . .

Why Numerical . . .

From Phenomena to . . .

Mathematical Modelling

Levels of Point of View

Numerical Treatment . . .

Related issues (What . . .

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Introduction to Scientific Computing

1. DefinitionMichael Bader


What is Scientific Computing?

October 22, 2003

http://www5.in.tum.de/~bader/


Why Numerical . . .






Literature

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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)



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2. Why Numerical Simulations?

2.1. since experiments are sometimes impossible:

• astrophysics: life cycle of galaxies etc.

• climate research: Gulf Stream, greenhouse effect etc.



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• weather forecast: tornadoes – where, when, and how strong?

• simulation of stockmarkets, or predicting economic effects (in-troducing the “Euro”)



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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



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2.3. since experiments can be extremely costly

• aerodynamics, turbulence: objects in a wind tunnel and so on

• car industry: crash tests



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• analysis and study of proteins

• Molecular dynamics: crystal structure, macromolecules



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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!



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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



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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



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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)



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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



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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.


introduction to scientific computing · what is scientiﬁc... why numerical... from phenomena...

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