Hit-and-Miss (or Rejection) Monte Carlo Method:

MM

M

M

MA

M largefor lim

4

bar

M

ii

tot

M

ii

OOOMM

MA

OOMM

1

1

1

(hit)

)1 (if otherwise (miss) 0

1 if (hit) 1

2

2

iii

iii

xyO

xyO

Hit-and-Miss (or Rejection) Monte Carlo Method:a “brute-force” method based on completely random

sampling

Then, how do we throw the stones and count them on computer?

:as observablean Define

times.Mfor ),( randomat Choose

O

yx ii

1. Generation of M 2D microstates

2. Measurement (Averaging an observable over M microstates)

x

y

A

O 1

1

(xi,yi)

1,0 ,1)( 2 xxxfy

Obtain an approximate value of A (= /4 M/M) by (1) throwing M stones in the square of the total area 1; (2) counting M stones in a quarter of the circle of the radius 1.

Lab 1. Calculation of the value of /4Simple, Brute-Force, Hit-and-Miss Monte Carlo

Method

Question 1 (Formulation of the problem = Restatement of the last page).

On computer, this problem can be formulated as obtaining the average (or expectation value) of an observable O over M instantaneous Oi values at each microstate (xi, yi) (i = 1 to M).

(1)What values can Oi take?

(2) How does the value of Oi depend on (xi, yi)?

bar

M

ii OOO

MM

M

M

MA

1

1

(throw)

(hit)

Obtain an approximate value of A (= /4 M/M) by (1) throwing M stones in the square of the total area 1; (2) counting M stones in a quarter of the circle of the radius 1.

Lab 1. Calculation of the value of /4Simple, Brute-Force, Hit-and-Miss Monte Carlo

Method

Question 1 (Formulation of the problem = Restatement of the last page).

On computer, this problem can be formulated as obtaining the average (or expectation value) of an observable O over M instantaneous Oi values at each microstate (xi, yi) (i = 1 to M).

(1)What values can Oi take?

Answer: 0 or 1

(2) How does the value of Oi depend on (xi, yi)?

Answer:

bar

M

ii OOO

MM

M

M

MA

1

1

(throw)

(hit)

21 if (hit) 1 iii xyO

)1 (if otherwise (miss) 0 2iii xyO

Lab 1. Calculation of the value of /4 (or )Question 2 (Algorithm & Flow chart).

Draw a flow chart to estimate A (or = 4A) by averaging over M instantaneous Oi values.

barestimated

bartot

M

ii

O

OOM

OO

MA

4

1

1

21 if ,1 iii xyO

otherwise ,0iO

Lab 1. Calculation of the value of /4

?1 2xy

Otot = Otot + O

O = 0.0

O = 1.0

yes

no

x=ran3(&seed)y=ran3(&seed)

i = 1

i = i + 1

?Mi no

Otot = 0.0

Obar = Otot/M; = 4xObar

yes

Question 2 (Algorithm & Flow chart).

Draw a flow chart to estimate A (or = 4A) by averaging over M instantaneous Oi values. 21 if ,1 iii xyO

otherwise ,0iO

Answer.

barestimated

bartot

M

ii

O

OOM

OO

MA

4

1

1

Question 3. Write a program to estimate A by averaging over M instantaneous O values. Try M = 10,000.

Lab 1. Calculation of the value of /4 (or )

trueOO

Flow chart for Question 4: Calculate .

21 if ,1 iii xyO

otherwise ,0iO

?1 2xy

Otot = Otot + O

O = 0.0

no

x=ran3(&seed); y=ran3(&seed)

i = 1

i = i + 1

?Mi no

Otot = 0; input seed

Obar = Otot/M; = 4Obar; true = acos(-1); = fabs(true )

yes

O = 1.0

yes

(accuracy)

error

1)]acos(

4

1

1

esttrue

esttrue

true

barestimated

bartot

M

ii

O

OOM

OO

MA

Question 3. Write a program to estimate A by averaging over M instantaneous O values. Try M = 10,000.


trueOO


Question 6. But, what if we don’t know the true value of what we estimate? Since we can’t estimate the error w.r.t. the true value, how could we estimate the accuracy (error) of the simulation?


(accuracy) or i.e., , deviation standard

)( variance

1limlim and

1

,,,,,, observablean of valuesousinstantane M ofset aFor

2222

11

21

OOOO

OOOO

OOM

OOOM

O

OOOOO

truetrue

true

M

iiMM

M

ii

Mi

Variance, Standard Deviation, Error, and Accuracy

Flow chart for Question 7: Calculate .

21 if ,1 iii xyO

otherwise ,0iO

?1 2xy

Otot = Otot + O; O2tot = O2

tot + O2

O = 0.0

O = 1.0yes

no


i = 1

i = i + 1

?Mi no

Otot = 0; O2tot =0; input seed

Obar = Otot/M; O2bar = O2

tot/M; = sqrt(O2bar

Obar2)

yes

4)( ,4

(accuracy)

deviation standard

11

)( variance

1

average

estimated

2

11

2

2222

1

O

OO

OO

OM

OM

OOOO

OM

O

true

true

M

ii

M

ii

M

ii

pi = 4 x Obar; _pi = 4 x


Answer: We expected that would be similar to ( ), but it’s not the case (

> ). (~constant with M) is much larger than (which decreases as M

increases)!Thus, itself is not a good indicator of the accuracy of a simulation.Question 8. Then, what else would be an indicator of the accuracy (error) of the

estimation,when we don’t know the true value of what we estimate?

?1 2xy

Otot = Otot + O; O2tot = O2

tot + O2

O = 0.0

O = 1.0yes

no


i = 1

i = i + 1

?Mi no

Otot = 0; O2tot =0

Obar = Otot/M; O2bar = O2

tot/M; = sqrt(O2bar

Obar2)

yes

pi = 4 x Obar; _pi = 4 x

one_estimate

Flow chart for Question 9:

Calculate over n estimates.

Flow chart for Question 9:

Calculate over n estimates.

Obartot = Obartot + Obar

Obarsqtot = Obarsqtot + Obar2

tot = tot +

one_estimate

j = 1

j = j + 1

?nj no

Obartot = 0; Obarsqtot =0; tot = 0; input seed

Obarbar = Obartot/n

Obarsqbar = Obarsqtot/n

bar= sqrt(Obarsqbar Obarbar

2)

bar = tot/n

yes

pi = 4xObarbar; barpi= 4xbar; barpi = 4xbar

4)( ,4)( ,4

1error of average

deviation standard

11

of variance

1

of average

, , , , , , i.e.

of estimatesn ofset aFor

estimated

1

2

1

2

1

222

1

21

O

n

On

On

OOO

On

OO

OOOO

O

n

jj

bar

n

jj

n

jj

n

jj

nj

Lab 1. Calculation of the value of

Lab 1. Calculation of the value of /4 (or )Question 10.But, what if we move onto a very complex problem? Each estimation (simulation) with M trials becomes so time-consuming that we couldn’t afford to running the simulation many times. What should we do in this case?

Answer: Let’s try to find a way to estimate the accuracy from one M-trial simulation.


deviation) standard trials,ofnumber ( MM

where,OOtrue

M largefor M

Hit-and-Miss (or Rejection) Monte Carlo Method:

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