Lecture Notes 1

What is Computational Science?What is Computational Science?Introduction to Computational Science , Fall 2010

Marcus PendergrassHampden-Sydney College

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

Outline

1 Basics

2 The Modeling Process

3 Types of Models

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 2 / 16

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

Computational Science

Definition

Computational science is the theory and practice of using modelingand simulation to solve problems.

ComputationalScience

ModelingSimulation

Problem Domain

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 3 / 16

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

Problem Domains

Definition

The problem domain is the field of study that the problem comesfrom.

Some important problem domains:

I biologyepidemiology, genomics, ecology, ...

I meteorologyweather prediction, climate modeling

I engineeringaircraft design, integrated circuit design, ...

I physicsgalaxy formation, stellar dynamics, ...

I many more...

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 4 / 16

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

Problem Domains

Definition

The problem domain is the field of study that the problem comesfrom.

Some important problem domains:

I biologyepidemiology, genomics, ecology, ...

I meteorologyweather prediction, climate modeling

I engineeringaircraft design, integrated circuit design, ...

I physicsgalaxy formation, stellar dynamics, ...

I many more...

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 4 / 16

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

Models

Definition

A model is a description of a system that reflects importantproperties of the system. A mathematical model is a model framed inthe language of mathematics.

Example

I System: a population whose rate of growth is proportional tothe size of the population.

I Mathematical model: dPdt = kP .

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 5 / 16

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

Models

Definition

A model is a description of a system that reflects importantproperties of the system. A mathematical model is a model framed inthe language of mathematics.

Example

I System: a population whose rate of growth is proportional tothe size of the population.

I Mathematical model: dPdt = kP .

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 5 / 16

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

Simulations

Definition

A simulation is physical system that mimics important properties ofsome other system. A computer simulation a simulation that isimplemented as a computer program.

Example

I System: a population whose rate of growth is proportional tothe size of the population.

I Mathematical model: dPdt = kP .

I Computer simulation: a Matlab program to solve themathematical model.

note:

Matlab is a computational environment for carrying out simulations.

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 6 / 16

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

Simulations

Definition

A simulation is physical system that mimics important properties ofsome other system. A computer simulation a simulation that isimplemented as a computer program.

Example

I System: a population whose rate of growth is proportional tothe size of the population.

I Mathematical model: dPdt = kP .

I Computer simulation: a Matlab program to solve themathematical model.

note:

Matlab is a computational environment for carrying out simulations.

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 6 / 16

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

Simulations

Definition

A simulation is physical system that mimics important properties ofsome other system. A computer simulation a simulation that isimplemented as a computer program.

Example

I System: a population whose rate of growth is proportional tothe size of the population.

I Mathematical model: dPdt = kP .

I Computer simulation: a Matlab program to solve themathematical model.

note:

Matlab is a computational environment for carrying out simulations.

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 6 / 16

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

Computational Environment

A computational environment is a computer program that facilitatescalculations. Computational environments may be fairly simple (e.g.graphing calculators), or very sophisticated. Examples ofcomputational environments include

I CalculatorsI calculator app on iPhone, TI-8x graphing calculators, etc.

I SpreadsheetsI Microsoft Excel, Apple Numbers, Open Office Calc

I Numerical environmentsI Matlab, Octave

I Computer algebra systemsI Maple, Mathematica, Sage

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 7 / 16

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

The Practice Of Computational Science

Interdisciplinary

Collaboration across disciplines is typical of computational science.

Teamwork

Teams usually include mathematicians, computer scientists, andexperts in the problem domain area (e.g. scientists, engineers).

Reporting

Results of computational science research and practice must becommunicated to a wider community in order to have an impact.

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 8 / 16

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

The Practice Of Computational Science

Interdisciplinary

Collaboration across disciplines is typical of computational science.

Teamwork

Teams usually include mathematicians, computer scientists, andexperts in the problem domain area (e.g. scientists, engineers).

Reporting

Results of computational science research and practice must becommunicated to a wider community in order to have an impact.

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 8 / 16

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

The Practice Of Computational Science

Interdisciplinary

Collaboration across disciplines is typical of computational science.

Teamwork

Teams usually include mathematicians, computer scientists, andexperts in the problem domain area (e.g. scientists, engineers).

Reporting

Results of computational science research and practice must becommunicated to a wider community in order to have an impact.

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 8 / 16

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

Outline

1 Basics

2 The Modeling Process

3 Types of Models

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 9 / 16

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

The Modeling Process

The basic modeling process

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 10 / 16

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

The Modeling Process

The basic modeling process

I Problems with verification and interpretation may cause you togo back to previous steps in the process.

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 11 / 16

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

The Modeling Process

The basic modeling process

I Problems with verification and interpretation may cause you togo back to previous steps in the process.

I Maintaining the model may also require you to revisit earliersteps.

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 12 / 16

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

Outline

1 Basics

2 The Modeling Process

3 Types of Models

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 13 / 16

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

Types Of Models

Deterministic and Stochastic Models

A probabilistic or stochastic model exhibits random effects. Adeterministic model does not. Many models are combinations of thetwo.

Static and Dynamic Models

A static model does not consider time, while a dynamic modelchanges with time.

Continuous and Discrete Models

A continuous-time model treats time as a continuous variable, able totake on any value. A discrete-time model treats time as a discretevariable, only able to take on certain pre-defined values.

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 14 / 16

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

Types Of Models

Deterministic and Stochastic Models

A probabilistic or stochastic model exhibits random effects. Adeterministic model does not. Many models are combinations of thetwo.

Static and Dynamic Models

A static model does not consider time, while a dynamic modelchanges with time.

Continuous and Discrete Models

A continuous-time model treats time as a continuous variable, able totake on any value. A discrete-time model treats time as a discretevariable, only able to take on certain pre-defined values.

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 14 / 16

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

Types Of Models

Deterministic and Stochastic Models

A probabilistic or stochastic model exhibits random effects. Adeterministic model does not. Many models are combinations of thetwo.

Static and Dynamic Models

A static model does not consider time, while a dynamic modelchanges with time.

Continuous and Discrete Models

A continuous-time model treats time as a continuous variable, able totake on any value. A discrete-time model treats time as a discretevariable, only able to take on certain pre-defined values.

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 14 / 16

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

Practice

Practice 1.1

What is the difference between a model and a simulation? Whichcomes first in the basic modeling process?

Practice 1.2

Suppose we want to model the angle of the minute hand of a clock(measured from the 12 o’clock position, say). If θ is the angle indegrees, and t is the time in minutes past the hour, find an equationthat models the relationship between θ and t.

Practice 1.3

Suppose I have a spreadsheet that tells me how much I have left topay on my mortgage at the end of each month. Does this represent adiscrete-time model or a continuous-time model?

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 15 / 16

Lecture Notes 1

Dr. Pendergrass

Basics

The ModelingProcess

Types of Models

Practice

Practice 1.4

Give an example of a system for which a probabilistic model would beappropriate. Give another example in which a probabilistic modelwould not be appropriate.

Practice 1.5

Write an essay describing an example of computational science inaction in industry, government, or academia. Carefully delineate theproblem that was addressed, and the computational science approachthat was used to solve the problem. Reference all sources used inyour report.

Dr. Pendergrass (H-SC) Lecture Notes 1 Fall 2010 16 / 16

BasicsThe Modeling ProcessTypes of Models