cds 130 - 003 fall, 2010 computing for scientists scientific simulation (sep. 27, 2010 – oct. 21,...

CDS 130 - 003Fall, 2010

Computing for Scientists

Scientific Simulation(Sep. 27, 2010 – Oct. 21, 2010)

Jie Zhang Copyright ©

Motivation• Scientific simulation allows one to reproduce the

details of a complex scientific system, e.g., physical systems, biological systems

• It is a new way of scientific research, in additional to traditional experimental and mathematical approaches.

ExampleMerger of the Milky Way: http://video.google.com/videoplay?docid=3671579993993423979#

The Milky Way and the Andromeda galaxy will likely fall together and merge within a few billion years. In this speculative simulation, the two galaxies flyby one another, exciting tidal tails and bridges and collide on a second pass finally merging after several convulsions. The last remnants of the smashed spirals show up as shells and ripples surrounding a newborn elliptical galaxy.

ExampleMany scientific simulations are based on so-called Navier-Stokes Equations (Note: do not worry about the math complexity)http://www.cfd-online.com/Wiki/Navier-Stokes_equations

Navier-Stokes Equations

ExampleCellular Blood Flow: http://www.youtube.com/watch?v=o11NDvrZMNs&feature=related

The RBCs are modeled as a membrane with hemoglobin inside with a viscosity of 6cP. Each RBC membrane is constructed of linear finite element triangular shells. These shell elements deform due to the fluid-structure interactions with the blood plasma and the hemoglobin.

Objectives

1. Mathematical Models

2. Numerical Methods• Iteration• Differentiation• Integration

3. Verification and Validation

A Pipeline of Models

1. Domain specialists (e.g., biologists) develop a conceptual representation of the system or a scientific model

2. The domain specialists collaborate with mathematicians to develop a mathematical representation, or mathematical model that corresponds to the science model

3. The domain specialists and mathematicians work with computational scientists to implement the equations and explore the results using a computer, that is to develop a computational model of the mathematical model.

4. The computational methods need to be verified, and the results need to be validated.

A common way that science gets done

Mathematical Models(Sep. 29, 2010)

Example 1

scientific model:

Every year your bank account balance increases by 20%

Mathematical model:

B(next-year) = B(this-year) + 0.2 * B(this-year)

Or

B(next-year) - B(this-year) = 0.2 * B(this-year)

Example 2scientific model:

Every year your bank account balance increases by 20%. Every year you pay a fee of $100 to the bank

Mathematical model:

B(next-year) = B(this-year) + 0.2 * B(this-year) - 100


Every year your bank account balance doubles

Mathematical model:

B(next-year) = 2 * B(this-year)


The death rate is 1% and no babies are born

Mathematical model:

P (next-year) = P(this-year) – 0.01*P(this-year)

OR: this year is represented by index I next year is represented by index 2

P(2) = P(1) – 0.01*P(1)

Example 5scientific model: The rabbit birth rate is 10% and no rabbits die

Mathematical model:

P (next-year) = P(this-year) + 0.1 * P(this-year)

Using index “i”, “i” represents this year. In the context of iteration, “I” represents current year

P(i+1) = P(i) + 0.1*P(i)

Example 5 (cont.)Mathematical model: P(i+1) = P(i) + 0.1*P(i)

One instance of simulation: Assuming initial population of 100, find the population in the next five year

• First year (i=1): P(1) = 100• Second year (i=2): P(2) = P(1) + 0.1*P(1) = 110• Third year (i=3): P(3) = P(2) + 0.1*P(2) = 121• Fourth year (i=4): P(4) = P(3) + 0.1*P(3) = 133• Fifth year (i=5): P(5) = P(4) + 0.1*P(4) = 146

Example 5 (cont.)

Calculation in Excel

>P(1) = 100>P(2) = P(1) + 0.1*P(1) = 110>P(3) = P(2) + 0.1*P(2) = 121>P(4) = P(3) + 0.1*P(3) = 133>P(5) = P(4) + 0.1*P(4) = 146

>plot(P,’*’)

Calculation in Matlab/Octav


The birth rate of rabbits is 10%. The death rate or rabbits is 0.02 times the number of rabbits multiplied by the number of foxes.

Mathematical model:

R (next-year) = R(this-year) + 0.10*R(this-year) –0.02*R(this-year)*F(this-year)

Use “i” represents the current year

R(i+1)=R(i) + 0.10*R(i) – 0.02*R(i)*F(i)

Example 7Scientific model:

The death rate of foxes is 10%. The birth rate of foxes is 2% of the number of rabbits multiplied by the number of foxes.

Mathematical model:

F (next-year) = ?

Example 8scientific model: • The birth rate of rabbits is 10%. The death rate or rabbits is

0.02 times the number of rabbits multiplied by the number of foxes.

• The death rate of foxes is 10%. The birth rate of foxes is 2% of the number of rabbits multiplied by the number of foxes.

Mathematical model:

R(i+1)=R(i) + 0.10*R(i) – 0.02*R(i)*F(i)

F(i+1)=F(i) - 0.10*R(i) + 0.02*R(i)*F(i)

Example 8 (cont.)Mathematical model: R(i+1)=R(i) + 0.10*R(i) – 0.02*R(i)*F(i)F(i+1)=F(i) - 0.10*F(i) + 0.02*R(i)*F(i)

One instance of simulation: Initial population of rabbits is 100, and that of foxes is 20. What are the population in the second year?

>R(1) = 100>F(1) = 50>R(2)=R(1) + 0.10*R(1) – 0.02*R(1)*F(1) = 70>F(2)=F(1) - 0.10*F(1) + 0.02*F(1)*R(1) = 58

Simulation Project

A closed system (e.g., Rabbits and Foxes on an island)•Change in number of rabbits per year

•increases in proportion to the number of rabbits (breeding like rabbits!) – the birth rate of prey•decreases in proportion to (the number of rabbits) x (number of foxes) - the death rate of prey due to interaction •Does not depend on rabbits dying of natural death

•Change in number of foxes per year •decreases in proportion to the number of foxes (more competition for food) – the death rate of predator•increases in proportion to (the number of rabbits) x (the number of foxes) – the birth rate of predator due to interaction

Predator – Prey Model

http://home.messiah.edu/~deroos/CSC171/PredPrey/PRED.htm

Simulation ProjectPredator – Prey Model

Simulation: Given (1) initial rabbit population, (2) initial fox population, (3) rabbit birth rate, (4) rabbit death rate due to interaction, (5) fox death rate, (5) fox birth rate due to interaction.

You need to find out how the population change with time?

Numerical Method - Iteration(Oct. 5, 2010)

Iteration

•Iteration is the most fundamental numerical method to solve the mathematical model that represents a dynamic scientific system, the evolution or the time change of the system.

• Used for solving differential equations•e.g., knowing the rate of change of the population

•Used for solving integration equations•e.g., total area enclosed by a Gaussian function

X(i+1) = X(i) + f(X(i))

2xey

Iteration

•The output of current iteration is used as the input of the next iteration

i: the sequence number of the iteration, representing the independent variable “time”, e.g., year in population growth

N: total number of iterations

X: the dependence variable, an unknown function, but the model seeks to find

f(X(i)): the known function representing the rate of change

>for i=1:N>X(i+1) = X(i) + f(X(i));>end

Example 1scientific model: The human population grows at an annual rate of 2%

Mathematical model:

P (i+1) = P(i) + 0.02 * P(i)

Question: In year 2000, the population is 6.0 billion. What are the populations in the next five years?

Work this out with the tools available to you: manual, calculator, excel, and program.

Example 1>P(1) = 6.0>for i=1:5>P(i+1)=P(i)+0.02*P(i);>end>P>plot(P,’*’)

Year 2000: 6.00Year 2001: 6.12Year 2002: 6.24Year 2003: 6.37Year 2004: 6.49Year 2005: 6.62

Example 1Question: In year 2000, the population is 6.0 billion. What

about the population in the next 200 years? What is the population in year 2200?

Work this out with the tools available to you: manual, calculator, excel, and program.

Example 1>P(1) = 6.0>for i=1:199>P(i+1)=P(i)+0.02*P(i);>end>P(199)>plot(P,’*’)>xlabel(‘Year from 2000’)>ylabel(‘Population (billion)’)>title(‘Population Growth’)>cd(‘~’)>print –dpng ‘population.png’

The answer is: the population is 302.60 billion in year 2200.

Example 1

Example 2Mathematical model of Predator-Prey scientific model R(i+1)=R(i) + 0.10*R(i) – 0.002*R(i)*F(i)F(i+1)=F(i) - 0.10*F(i) + 0.002*R(i)*F(i)

Question: Initial population of rabbits is 100, and that of foxes is 10. What are their populations in the next 10 years?

>R(1) = 100>F(1) = 10>R(2)=R(1) + 0.10*R(1) – 0.002*R(1)*F(1) = 108>F(2)=F(1) - 0.10*F(1) + 0.002*F(1)*R(1) = 11>R(3)=…>F(3)=…>……..

Example 2>R(1) = 100>F(1)=10>for i=1:10>R(i+1)=R(i)+0.10*R(i)-0.002*R(i)*F(i);>F(i+1)=F(i)-0.10*F(i)+0.002*R(i)*F(i);>end>R>F>plot((R)>hold >plot(F)>print –dpng ‘predator_prey.png’

Example 2

Differential Equations(Oct. 7, 2010)

Differential Equations

Differential equations play an important role in modelling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons(http://en.wikipedia.org/wiki/Differential_equation)

Differential Equations •A differential equation is a mathematical equation for an unknown function of one or several variables that relates its derivatives and the function

•It is used to describe a dynamic system, in which the time rate of change is well defined

Example 1scientific model: The human population grows at an annual rate of 2%

Pdt

dP02.0

-The rate of change is known-The population evolution is unknown, but described by the

following differential equation

dP/dt: the rate of change of the populationP: the unknown function P, a dependent variable of t t: time, the independent variable

dP: differential of P, or change of Pdt: differential of t, or change of t

Example 1

small infinitely togoes as

change) of rate time( timeof change

population of change

t

t

P

dt

dP

• Δt = 10 years, only know population change every 10 years

• Δt = 1 year, only know population change every 1 years• Δt = 1 day, can calculate population change on daily basis

The smaller the time interval, the more accurate the result.

Example 1

P(i)*0.02P(i)1)P(i

or

P(i)*0.02P(i)-1)P(i

Thus

)()1()()1(

)()1(

02.0

iPiPii

iPiP

t

P

dt

dP

Pdt

dP

Differentiation to Iteration

Example 1Iteration to Differentiation

Pdt

dP

Thereforedt

dP

t

P

t

P

ii

iPiP

02.0

,

)()1(

)()1(P(i)-1)P(i

Further

P(i)*0.02P(i)-1)P(i

P(i)*0.02P(i)1)P(i

Example 2

Newton’s Law of Cooling:

minuteper ofunit in t,coefficien cooling :q

minute ofunit in time,:t

mperatureambient te :T

object of ure temperat:T

)(*

a

aTTqdt

dT

A scientific law is often expressed in differential equations, for good reasons

The cooling rate of an object is proportional to the temperature difference between the object and the ambient temperature

Example 2Question: An iron bar is heated to 1000°C. Assume the

cooling rate is 0.2 min-1, and the ambient temperature is 30°C. At approximately what time (in unit of minute) does the temperature equal one-half of the initial temperature after the iron bar is allowed to cool?

Example 2

Convert differential equation to iteration

))((*T(i)1)T(i

or

))((*T(i)-1)T(i

Thus

)()1()()1(

)()1(

)(*

a

a

a

TiTq

TiTq

iTiTii

iTiT

t

T

dt

dT

TTqdt

dT

Example 2>cd(‘~’)>edit cooling.m>cooling>print –dpng ‘cooling.png’

function [T] = cooling ()

T(1)=1000 % initial temperatureTa=30 % ambient temperatureq = 0.2 % cooling coefficient

%guess the iteration needed%N = 1000 % iteration 1000 times, or calculate up to 1000 minutesN=50 %second guess of appropriate iteration number

%start the iteration calculation using for loopsfor i=1:N T(i+1)=T(i)-q*(T(i)-Ta);endfor

plot(T,'*')xlabel('Time (min)')ylabel('Temperature (deg)')title('cooling curve')

endfunction

“cooling.m”

Example 2

Answer: It takes about 4.5 minutes to cool down to the half temperature

Example 3Mathematical model of Predator-Prey scientific model R(i+1)=R(i) + 0.10*R(i) – 0.002*R(i)*F(i)F(i+1)=F(i) - 0.10*F(i) + 0.002*R(i)*F(i)

FRFdt

dF

FRRdt

dR

**002.0*1.0

**002.0*1.0

Mathematical notation in differential equations:

Example 3

Question: Initial population of rabbits is 100, and that of foxes is 10. What are their populations in the next 50 years?

1. Use a program file to save all the code2. Use variables to represent the coefficients3. run the program file

Differential Equations (continued):

Interval and Sub-interval

(Oct. 14, 2010)

Interval and Sub-interval •In numerical simulation, one has to specify the calculation interval and sub-interval•Interval is the domain of the calculation

•In the population example, if you are asked to calculate the population in the next 10 years, what is the interval or domain of the calculation

•Answer: from year 1 to year 10, or i=[1:10] in the iteration format•Answer: time t in [1,10], or from 1 to 10 in the differentiation format

Population Example:

P(i)*0.02P(i)1)P(i

or ,02.0

Pdt

dP

Sub-interval •Sub-interval is the step-size in each iteration of the numerical calculation

•In the population example, so far we have always assumed that the sub-interval is 1 year

•That is, from i to i+1, the time interval is one year•However, one can choose a different sub-interval, for instance, one month

•This can be used to study the population change on a monthly base

•In the differential notation, dt is formerly defined as infinitely small time interval

Population Example:P(i)*0.02P(i)1)P(i

or ,02.0

Pdt

dP

P(i)*0.02P(i)1)P(i

or

P(i)*0.02P(i)-1)P(i

Thus

)()1()()1(

)()1(

02.0

iPiPii

iPiP

t

Pt

P

dt

dP

Pdt

dP

Differentiation Equation to Iteration Equation

Sub-interval

In the conversion, we’ve made the following approximations

(1) Differentiation dt is replaced by finite difference Δt (the time step, or the sub-interval); Correspondingly, dP replaced by ΔP

(2) Assuming the sub-interval to be 1; in this example, means one year

Question to think?

What if you want to study the population change on a finer time resolution, say from month to month?

Sub-interval

•You have to change the step size: Δt = 1/12 year, or one month•One iteration from i to i+1 is now one month, instead of one year •The rate of change of the population is now (0.02/12) for one month

P(i)*0.02/12)(P(i)1)P(i

Example

>P(1)=6.0>for i=[1:2]>P(i+1)=P(i)+0.2*P(i);>end

Δt = 1 (year)two iteration

Δt = 1/12 (year)24 iteration

Question: Given initial population of 6.0 billion, what is the population in the beginning of the third year?

P = 6.2424

>P(1)=6.0>for i=[1:24]>P(i+1)=P(i)+(0.2/12)*P(i);>end

P = 6.2447

The two results are close, but different!

The Effects of Sub-interval •Smaller sub-interval requires more steps of calculation

•But, smaller sub-interval provides more detailed evolution

•Further, smaller sub-interval provides more accurate results, e.g., fully illustrated in the context of integration

Integration(Oct. 14, 2010)(Oct. 19, 2010)

Integration •Integration is like the opposite operation of the differentiation

•Differentiation: the rate of change of a function,•e.g. given what is f(i+1)-f(i)

•Integration: the total sum of a function over a domain•e.g., find what is f(i+1)+f(i)

Primitive Integration •Primitive integration method has long been used by ancient mathematicians•A good example is to find the area size of a circle, that leads to the discovery of π

•The piece-wise approach•Count the small tiles with known size•The smaller the tile, the more accurate the circular area, thus the π•In other words, the smaller the sub-interval, the more the accuracy

S = π r2

Mathematical notation

b

a

dxxfF )(

•f(x): the integrant, the function of concern•F: the total value of integration•[a, b]: the interval of integration

• a: lower limit, b: upper limit•dx: the sub-interval of integration. Infinitely small in calculus

Integration: geometric meaning

Integration can be defined as the total area enclosed by the function between the integration interval

Integration:Piece-wise Approach

The total area is approximated by the sum of the sizes of the rectangles (L1+L2+L3….). The size of each rectangle can be easily calculated over the sub-interval: Li = f(xi) * Δx

Integration to Iteration

xxfiFiF

x

x

N

xxfxxfxxfxxfF

xfxf

xdx

dxxfF

i

N

iiN

i

b

a

)()()1(

1)-(ixx

a)/N-(b interval-sub :

iterations ofnumber or ,rectangles ofnumber :

)()(....)()(

)()(

)(

1i

121

Example: f(x)=x

5

0

)(

xdxF

xxf

Question: integrate the function f(x)=x from the interval x=0 to x=5.

It is equivalent to find the area of the triangle with the base of 5 and the height of 5. The area is expected to be 12.5

Example: f(x)=x

101)43210(

54321

445

334

223

112

;01

0.15/)00.5(/)(

F

xxxxxxxxxxF

xxx

xxx

xxx

xxx

ax

Nabx

Using 5 rectangles: N=5,

The approximate value is smaller than the true value

Example: f(x)=xUsing 10 rectangles: N=10, sub-interval Δx=0.5

N=10; %number of rectangles, or iterationa=0; % lower limitb=5.0; % upper limitdel_x=(b-a)/N; %the sub-interval

x1=a;F(1)=x1*del_x;

for i=[2:N] xi=x1+(i-1)*del_x; F(i)=F(i-1)+xi*del_x;end

“area1.m”

F=11.25

Example: f(x)=xUsing 20 rectangles: N=20, sub-interval Δx=0.25

F=11.875

Using 50 rectangles: N=50, sub-interval Δx=0.10

F=12.25


F=12.475


F=12.4975 Therefor, the smaller the sub-interval, the better the accuracy

Example: f(x)=x2

Question: integrate the function f(x)=x2 from the interval x=0 to x=5. Find the value when the number of sub-intervals is chosen to be (1)N=5(2)N=10(3)N=100(4)N=1000

5

0

2

2)(

dxxF

xxf

Example: f(x)=x2

301)169410(

54321

445

334

223

112

;01

0.15/)00.5(/)(

22222

F

xxxxxxxxxxF

xxx

xxx

xxx

xxx

ax

Nabx

Using 5 rectangles: N=5,

This approximate value is smaller than the true value

IntegrationQuestion: How does the integration accuracy depend on the size of the sub-interval? Why?

Verification and Validation(Oct. 19, 2010)

Scientific Method

•Four essential elements of a scientific method:1.Characterization: observation and measurement2.Hypothesis3.Prediction4.Testing

•It is an iterative process until the prediction is consistent with the observation•The hypothesis becomes a theory.

Verification and validation is an integral part of the scientific method.

Verification Versus Validation For the scientific method that involves scientific simulation, the testing has two basic stages:1.Verification: Are you solving the equations right?2.Validation: Are you solving the right equation?

•Verification is to make sure that your computer program is correct •Validation is to make sure that your scientific hypothesis, or the scientific model, is correct.

•A scientific model is often specified by1. Equations2. Adjustable parameters, e.g., growth rate

Example We use a study on flu outbreak to demonstrate the scientific method and the validation process.

Data are based on the article http://dx.doi.org/10.1136/bmj.1.6112.586

In a school of 762 students, the first flu case was reported on Jan. 22. There were 6 cases found on Jan. 23. The number of infected students grew to 26, 73 and 222 in the following three days. Construct a scientific model to explain the outbreak?

Characterization (Step 1)

•Observe the phenomena•Collect the data•Analyze the data

Hypothesis (Step 2)

Forming the hypothesis

Model 1

The number of students infected equals 10 times the day number since the outbreak, with day number = 1 corresponding to the day the first student was infected.

Prediction (Step 3)

•Formulation of a predictive test•Create a mathematic model

•Implement and run the computational model•Obtain the predictive result

iiN 10)1(

Testing (Step 4)

•Verify that the calculation is done right•Do sanity checks: Does the simulation predict behavior that would never happen in reality? Does population ever go negative? Does the mass stay positive? •Compare the model prediction with existing data•Determine if the prediction “matches” the observation

•Apparently, the match is “not good”•How to determine “not good” or “good”. This will be explored in the next chapter.

Model 2: Repeat step 2, 3 and 4 with different scientific model

Hypothesis (Step 2)


Model 2

The number of students infected equals 20 times the day number since the outbreak, with day number = 1 corresponding to the day the first student was infected.

Prediction (Step 3)



iiN 20)1(

Testing (Step 4)

•Again, the model prediction does not fit the observed data•Need a better model

•Need to further study the data•Need to think about the scientific justification of the model

Model 3: Repeat step 2, 3 and 4 with different scientific model

Hypothesis (Step 2)


Model 3

The number of new people infected on a given day is proportional to the number of people infected on the previous day. The proportion rate is an adjustable parameter. In this model, it is assumed to be 3.

Prediction (Step 3)



)()()1(

0.3

iNaiNiN

a

Testing (Step 4)

•This model is “good”•It is a good representation of the observation

•Caveats•It is not an exact representation, but a good approximation of the observations•It is only validated on the initial stage of an outbreak. It is not a model to explain the long term evolution.

Scientific Method It is an iterative process. One needs to re-form the hypothesis and re-test the result.

The end(Oct. 21, 2010)

cds 130 - 003 fall, 2010 computing for scientists scientific simulation (sep. 27, 2010 – oct. 21,...

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