Description
Resumo
In this class you will learn/remember some basic concepts of concurrency and how to process data in parallel using the C programming language.
1 Introduction
The “Monte Carlo Method” is a method of solving problems using sta- tistics. Given the probability, p, that an event will occur in certain conditions, a computer can be used to generate those conditions repe- atedly. The number of times the event occurs divided by the number of times the conditions are generated should be approximately equal to p.
Figure 1 shows a circle with radius r = 1 inscribed within a square.
The area of the circle is πr2 = π12 = π, and the area of the square is
(2r)2 = 22 = 4. The ratio of the area of the circle to the area of the
square is p = Area of circle = π = 0.7853981634 Area of square 4
Figura 1: A circle within a square.
|
r=1 |
If we could compute ratio p, then we could multiple it by four to
obtain the value π = p × 4. One particularly simple way to do this is to pick lattice points in the square and count how many of them lie inside the circle (see Figure2). Suppose for
example that the points 2i−1 , 2j−1 32, 32 are selected, then there are 812 points inside 32 32 i=1,j=1
the circle and 212 points outside the circle and the percentage of points inside the circle is p = 812 = 812 = 0.792195122. Then the area of the circle is approximated with the
following calculation: Area of circle = p × Area of square = p × 4 = 0.792195122 × 4 = 3, 168780488.
812+212 1024
|
r=1 |
2 Monte Carlo Method for π
Monte Carlo methods can be thought of as statistical simulation methods that utilize a sequences of random numbers to perform the simulation. The name “Monte Carlo” was coined by Nicholas Cons- tantine Metropolis (1915-1999) and inspired by Stanslaw Ulam (1909- 1986), because of the similarity of statistical simulation to games of chance, and because Monte Carlo is a center for gambling and games of chance. In a typical process one compute the number of points in a set A that lies inside box R. The ratio of the number of points that
Figura 2: Ratio between areas of circle and square.
fall inside A to the total number of points tried is equal to the ratio of the two areas (or volume in 3 dimensions). The accuracy of the ratio p depends on the number of points used, with more points leading to a more accurate value.
A simple Monte Carlo simulation to approximate the value of π
could involve randomly selecting points (xi,yi)ni=1 in the unit square
and determining the ratio p = mn where m is number of points that
satisfy x2i + yi2 ≤ 1. In a typical simulation of sample size n = 1000
there are 787 points satisfying that equation. Using this data we obtain
p=m = 787 =0.787andπ=p×4=0.787×4=3.148. n 1000
Every time a Monte Carlo simulation is made using the same sam- ple size n it will come up with a slightly different value. The values converge very slowly of the order O(n−1/2). This property is a conse- quence of the Central Limit Theorem.
You may find a web simulation of this method at https:// academo.org/demos/estimating-pi-monte-carlo/.
3 Lab Work
3.1 Sequential Version
Design and implement a C program named approxPi that approximates the value of π by using the Monte Carlo method. The program must receive a command line argument that specifies the number of simulations to be executed (i.e., the number of points to be generated) and provide an output as given in the example below. Try it it multiple values for the number of simulations, e.g.,
$ approxPi 1000 Total Number of points: 1000 Points within circle: 779 Pi estimation: 3.11600
$ approxPi 100000 Total Number of points: 100000 Points within circle: 78656 Pi estimation: 3.14624
You may have a goos estimation of the time your program takes to run the full simu- lation by using the command time, e.g.,
$ time approxPi 1000 Total Number of points: 1000 Points within circle: 779 Pi estimation: 3.11600
real 0m0.007s user 0m0.001s sys 0m0.003s
r=1
2
Figura 3: Monte Carlo method.
Use this simple lab exercise to learn how to use GIT. Create a repository in a free public server (e.g., github, gitlab, bitbucket, …) and use it to manage the versioning of your code.
3.2 Parallel Version
Now, duplicate the source code of your original (sequential) C program and develop a parallel version approxPiPar using pthreads. This parallel version shall accepts a second argument (optional, defaults to one) indicating how many parallel threads shall be execu- ting.
Remember to keep on using GIT. 🙂
3.3 Experimental evaluation
Add a flag to your command line arguments that will make your program output these plain values separated by tabs: n_iterations, n_threads, run_time. Now, run your program with different values for iterations and processors. Also, remember to run each configuration at least three times. You may use the script below to help you run the tests:
ITEREATIONS="1000 10000 100000 1000000 10000000 100000000" THREADS="1 2 4 8 16" NRUNS="5" for IT in $ITEREATIONS; do
for TH in $THREADS; do
for R in ($‘$seq 1 $NRUNS); do
approxPi $ITEREATIONS $THREADS
done
done done | tee output_file.csv
name it
file_name.sh
and run it with
bash file_name.sh
3.4 To Think About. . .
The sequential version is faster, slower or identical to the parallel version with just one thread?
The parallel version with two thread is faster than with one? When you double the number of threads does it take half the time? More? Less?
Which of your implementations is faster? C or Java? Why?
3
Final Note
If you search the web, it will be trivial to find an implementation of the Monte Carlo simulation to approximate the value of π, that you can easily adapt to your needs. But note that if you do that, you are cheating and you will not learn what you are supposed to learn in this lab class. Just be honest to yourself and make your own program. 😉
Acknowledgments
The text from the first two sections is an adaptation from the text in http://mathfaculty. fullerton.edu/mathews/n2003/montecarlopimod.html.
