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COMPUTATIONAL MODELLING and

SIMULATIONModelling and Scientific Computing

Research Group

Speakers: H. Ruskin/L.Tuohey

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INTRODUCTION

• HISTORY: 1953

MANIAC simulates liquid !

• Monte Carlo influence in Los Alamos

• Dynamic developments 1957,1959…..

…..Computational Science…..!

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CM &S. - Motivation

• EXACTLY soluble/Analytical Methods• IF NOT? …. Approximate?• Simulation “Essentially” exact

“Essence” of problem

copes with intractability

can “test” theories and experiment

direct route micro macro

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CM &S. - applications

• Examples

Atoms … to galaxies, polymers, artificial life, brain and cognition, financial markets and risk,traffic flow and transportation,

ecological competition, environmental hazards,……….

Nonlinear, Non-equilibrium...

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EXPERIMENT, THEORY & COMPUTER SIMULATION

E X P E R IM E N TA LR E S U L TS

E X P E R IM E N T

R E A L W O R L D B U IL D P H Y S IC A LM O D E L

TE S TS O FM O D E L S

TE S TS O FTH E O R IE S

C O M P A R E

E X A C T R E S U L TSF O R M O D E L

S IM U L A TIO N

TH E O R E TIC A L P R E D IC TIO N S

C O N S TR U C T A P P R O X .TH E O R IE S

TH E O R E TIC A LM O D E L

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CORE ACTIVITIES "PHASES" -in aCOMPUTATIONAL SCIENCE Investigation

Development of the maths. model Development of the algorithm for

numeric solution Implementation of the algorithm Generation of solutions e.g.

through numerical simulation ofthe scientific phenomenon

Representation of the computedresults e.g. through graphicalvisualization

Interpretation of results ( with adegree of "healthy scepticism")and model validation

KEY ASPECTS

of

SCIENTIFICCOMPUTING

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COMPLEX SYSTEMS

• Size - billions of elements, events etc. many variables,

• Changes - dynamics

• Lack of Sequence or pattern

• Instability - (Non-equilibrium)

• Non-constant cause-effect (Non-linearity)

• Global vs local changes

etc.

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COMPLEXITY 2 - WHAT SORT OF TOOLS?

• Cellular Automata- “chess-board”

• Monte Carlo- random numbers

• Lattice Gas Models - Lattice-based, conservation laws “Fluid” Models

• SOC - self-driven catastrophes

• Neural Networks - content addressable memory

• Genetic Algorithms- model evolution by natural selection; mutation, selection

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MODELS - why do we need them?

• TO PICTURE HOW SYSTEM WORKS

• TO REMOVE NON-ESSENTIALS , called reducing the “degrees of freedom” (simple model first)

• TO TEST IT

• TO TRY SOMETHING DIFFERENT

…….cheaply and quickly

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CATEGORIES of COMPUTATION

• numerical analysis (simplification prior to computation)

• symbolic manipulation (mathematical forms e.g. differentiation, integration, matrix algebra etc.)

• simulation (essential elements - minimum of pre- analysis)

• data collection/analysis• visualisation

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SIMULATION LEVELS

• BRIDGING KNOWLEDGE GAP

Idealised Model Algorithm Results

• GOALS - SIMPLE LAWS

- PLAUSIBILITY

- MEW METHODS/MODELS

• DIRECT / INDIRECT

• PHENOMENOLOGY vs DETAIL

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NONLINEAR SYSTEMS

• MOST Natural Phenomena - Nonlinear

e.g. Weather patterns, turbulent flows of liquids, ecological systems = geometrical

• CHAOS

e.g. unbounded growth or population explosion cannot continue indefinitely - LIMIT = sustainable environment

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NON-EQUILIBRIUM SYSTEMS

• Inherently UNSTABLE

• FEW THEORIES - often simulation leads the way

• EXAMPLE - froth coarsening

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HOW FROTHS EVOLVE

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EXAMPLE- Financial Markets, volatility/ turbulence

Sequence number

1654

1567

1480

1393

1306

1219

1132

1045

958

871

784

697

610

523

436

349

262

175

88

1

Te

chn

ica

l1400

1200

1000

800

600

400

200

0

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- Finance 2

• KEY EVENTS 07/97 - 11/97 Roller-Coaster Asian Crisis

14/09/98 -South American BAD news

23/07/99 -plunge after highs/Greenspan address

12/12/00 -U.S. Supreme Court judgement on Election Result

28/03/01 -cut in Fed. Reserve rate not enough

18/09/01 -post Sept. 11th

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EXAMPLEImmunological Response

Antigens

Macrophage (M)

Helper (H)

B cells

ViralInfectedcell(V)

Cytotoxic Killer cells(C)

Cell mediated arm of Immune-response

Humoral arm .. etc.

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CURRENT- Immunology2

• Physical Space – Stochastic C.A– Microscopic– Macroscopic

• Shape Space– N-Dimensional Space that models affinity rules

and repertoire size

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SHAPE SPACE FORMULATION

V

V

V

V

R

introduced to predict repertoire size

Sphere of Influence for Immune System Components

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ALGORITHMS - MD, MC etc.

• MD - many particle - build dynamics from known interactions

• MC - most probable outcome (random numbers)

• CA - discrete dynamical systems. Finite states. Local updates.

• FD/BV - solns. D.E.’s - e,g. predictor/ corrector algorithms

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Example - TRAFFIC FLOW as a C.A. Model

c0 0 0 b b

0 a a

Major-stream

AVMinor-stream

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Algorithms - contd

• PROBLEM - open-ended

• CORRECT/ENOUGH?

Basics - compare with known results

- Orders of Magnitude

- Errors/Limits

- Extensions?

Coherent Story

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LANGUAGES ETC.

• PROCEDURAL/ FUNCTIONAL, OBJECT-ORIENTED - Fortran , C (change state or memory of machine by sequence of statements); LISP, Mathematica, Maple (function takes I/P to give O/P); C++, JAVA (Program = structured collection of objects)

• PLATFORM

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NUMBERS, PRETTY PICTURES and INSIGHT

• NUMBERS, PICTURES AGREEMENT? - not enough

e.g. Simulation of river networks as a Random Walk. Path of Walker = Meandering of River

……..Why?