Introduction to Scientific Computing · 2006-04-20 · Introduction to Scientiﬁc Computing 1....

What is Scientific . . .

What’s in a Name?

Why Numerical . . .

Grand Challenges

From Phenomena to . . .

Mathematical Modelling

Levels of Point of View

What else has to be . . .

Numerical Treatment . . .


Literature

Page 1 of 20

Introduction to Scientific Computing

1. DefinitionMiriam Mehl


What is Scientific Computing?

October 28, 2002

http://www5.in.tum.de/persons/mehl/


What’s in a Name?

Why Numerical . . .

Grand Challenges







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 concrete task of simulation:

– reconstructand understandknown scenarios (natural disas-ters)

– optimizeknown scenarios (technical processes)

– predictunknown scenarios (weather, new materials)



What’s in a Name?

Why Numerical . . .

Grand Challenges







Literature

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2. What’s in a Name?

• Scientific computing?

• scientific and engineering computing?

• computational science and engineering?

• simulation?

A chemist’s provoking question:

What the hell is nonscientific computing?

The scientific computer’s answer:

What you do, for example!



What’s in a Name?

Why Numerical . . .

Grand Challenges







Literature

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3. Why Numerical Simulations?

3.1. since experiments are sometimes impossible:

• astrophysics: life cycle of galaxies etc.

• geophysics: displacement of the earth’s magnetic field

(movie)



What’s in a Name?

Why Numerical . . .

Grand Challenges







Literature

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• climate research: Gulf Stream, greenhouse effect etc.

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



What’s in a Name?

Why Numerical . . .

Grand Challenges







Literature

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• security: 1993 bomb attack in the World Trade Center

• propagation of harmful substances

• economics: development of the stock market etc.

• medicine: adaptive materials in implants



What’s in a Name?

Why Numerical . . .

Grand Challenges







Literature

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3.2. since experiments are sometimes very unwelcome

• the bad guy: tests of nuclear weapons

• statics: stability of buildings etc.



What’s in a Name?

Why Numerical . . .

Grand Challenges







Literature

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3.3. since experiments are sometimes rather unwelcome

• natural disasters: avalanches

• security: effects of car bombs



What’s in a Name?

Why Numerical . . .

Grand Challenges







Literature

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3.4. since experiments are sometimes extremely costly

• influence of radiation on the genetic makeup

• analysis and study of proteins



What’s in a Name?

Why Numerical . . .

Grand Challenges







Literature

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• Molecular dynamics: crystal structure, macromolecules



What’s in a Name?

Why Numerical . . .

Grand Challenges







Literature

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3.5. since simulations are sometimes just cheaper or faster,resp.

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

• process engineering: stirring and mixing processes



What’s in a Name?

Why Numerical . . .

Grand Challenges







Literature

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• car industry: vehicle dynamics, elk test

• car industry: crash tests



What’s in a Name?

Why Numerical . . .

Grand Challenges







Literature

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4. US Government: Grand Challenges

• climate research

• combustion

• automobile development

• aircraft design

• electronic design automation

• biology and medicine

• chemistry and physics

• material science

• financial engineering

• . . .



What’s in a Name?

Why Numerical . . .

Grand Challenges







Literature

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5. 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



What’s in a Name?

Why Numerical . . .

Grand Challenges







Literature

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6. Mathematical Modelling

model as a (simplifying) formal abstraction of reality

issues when deriving a mathematical model:

– Which quantities have some influence, and how importantis it?

– What relations exist between them? "Which type of mathe-matics?"

– What is the given task (solve, optimize, etc.)?

issues when analyzing a mathematical model:

– What can be said about existence and uniqueness of solu-tions?

– 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

example: simulations concerning man



What’s in a Name?

Why Numerical . . .

Grand Challenges







Literature

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7. 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’s in a Name?

Why Numerical . . .

Grand Challenges







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8. What else has to be done?

In practice, models can typically not be solved analytically

numerical (approximate and computer-based) methods!

the numerical part is non-trivial:

– often complicated geometries (seeping processes in soil)

– often changing geometries (a sail in the wind)

– accuracy requirements force a high resolution of the do-main

– also higher dimensional problems:

– quantum mechanics: d=6,9,12,... , finance: d=365

– time dependence (unsteady phenomena)

– high memory requirements

– often poor convergence of standard methods (long run times)

– multiscale phenomena (turbulent flows)



What’s in a Name?

Why Numerical . . .

Grand Challenges







Literature

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9. Numerical Treatment of Models

many approximations and compromises:

numbers fixed number of digits instead of real numbers

functions approximating polynomials instead of series

domains polygonally bounded, restriction to grid points

operators difference quotients instead of derivatives

function spacesonly finite-dimensional

requirements to be fulfilled by numerical algorithms:

efficient high accuracy with moderate storage investment

fast the approximate solution is computed in short time

stable no significant/qualitative errors in the results

robust can be applied for a large class of problems

main tasks:

– derive the discretized equations

– solve the resulting discrete system of equations



What’s in a Name?

Why Numerical . . .

Grand Challenges







Literature

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10. 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’s in a Name?

Why Numerical . . .

Grand Challenges







Literature

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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.


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Introduction to Scientific Computing · 2006-04-20 · Introduction to Scientiﬁc Computing 1....

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