Monte Carlo Simulation

• Used when it is infeasible or impossible to compute an exact result with a deterministic algorithm

• Especially useful in – Studying systems with a large number of coupled degrees of

freedom• Fluids, disordered materials, strongly coupled solids, cellular

structures

– For modeling phenomena with significant uncertainty in inputs

• The calculation of risk in business

– These methods are also widely used in mathematics • The evaluation of definite integrals, particularly multidimensional

integrals with complicated boundary conditions

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Monte Carlo Simulation

• No single approach, multitude of different methods

• Usually follows pattern– Define a domain of possible inputs – Generate inputs randomly from the domain – Perform a deterministic computation using the

inputs – Aggregate the results of the individual

computations into the final result • Example: calculate Pi

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3

Monte Carlo: Algorithm for Pi• The value of PI can be calculated in a number of

ways. Consider the following method of approximating PI Inscribe a circle in a square

• Randomly generate points in the square • Determine the number of points in the square that

are also in the circle • Let r be the number of points in the circle divided

by the number of points in the square • PI ~ 4 r • Note that the more points generated, the better

the approximation • Algorithm :

npoints = 10000

circle_count = 0

do j = 1,npoints

generate 2 random numbers between 0 and 1

xcoordinate = random1 ; ycoordinate = random2

if (xcoordinate, ycoordinate) inside circle

then circle_count = circle_count + 1

end do

PI = 4.0*circle_count/npoints

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5

OpenMP Pi CalculationInitialize variables

Initialize OpenMP parallel environment

Calculate PI

Print value of pi

N WorkerThreadsMaster Thread

Generate random X,Y Generate random X,Y Generate random X,Y

Calculate Z=X^2+Y^2 Calculate Z =X^2+Y^2

If point lies within the circle

Calculate Z =X^2+Y^2

If point lies within the circle

If point lies within the circle

Count ++ Count ++Count ++

Reduction ∑

Y

N N N

Y Y

OpenMP Calculating Pi

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#include <omp.h>#include <stdlib.h>#include <stdio.h>#include <time.h>#define SEED 42

main(int argc, char* argv){ int niter=0; double x,y; int i,tid,count=0; /* # of points in the 1st quadrant of unit circle*/ double z; double pi; time_t rawtime; struct tm * timeinfo;

printf("Enter the number of iterations used to estimate pi: "); scanf("%d",&niter); time ( &rawtime ); timeinfo = localtime ( &rawtime );

Seed for generating random number

http://www.umsl.edu/~siegelj/cs4790/openmp/pimonti_omp.c.HTML

OpenMP Calculating Pi

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printf ( "The current date/time is: %s", asctime (timeinfo) ); /* initialize random numbers */ srand(SEED);#pragma omp parallel for private(x,y,z,tid) reduction(+:count) for ( i=0; i<niter; i++) { x = (double)rand()/RAND_MAX; y = (double)rand()/RAND_MAX; z = (x*x+y*y); if (z<=1) count++; if (i==(niter/6)-1) { tid = omp_get_thread_num(); printf(" thread %i just did iteration %i the count is %i\n",tid,i,count); } if (i==(niter/3)-1) { tid = omp_get_thread_num(); printf(" thread %i just did iteration %i the count is %i\n",tid,i,count); } if (i==(niter/2)-1) { tid = omp_get_thread_num(); printf(" thread %i just did iteration %i the count is %i\n",tid,i,count); } http://www.umsl.edu/~siegelj/cs4790/openmp/pimonti_omp.c.HTML

Initialize random number generator; srand is used to seed the random number generated by rand()

Randomly generate x,y points

Initialize OpenMP parallel for with reduction(∑)

Calculate x^2+y^2 and check if it lies within the circle; if yes then increment count

Calculating Pi

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if (i==(2*niter/3)-1) { tid = omp_get_thread_num(); printf(" thread %i just did iteration %i the count is %i\n",tid,i,count); } if (i==(5*niter/6)-1) { tid = omp_get_thread_num(); printf(" thread %i just did iteration %i the count is %i\n",tid,i,count); } if (i==niter-1) { tid = omp_get_thread_num(); printf(" thread %i just did iteration %i the count is %i\n",tid,i,count); } } time ( &rawtime ); timeinfo = localtime ( &rawtime ); printf ( "The current date/time is: %s", asctime (timeinfo) ); printf(" the total count is %i\n",count); pi=(double)count/niter*4; printf("# of trials= %d , estimate of pi is %g \n",niter,pi); return 0;}

http://www.umsl.edu/~siegelj/cs4790/openmp/pimonti_omp.c.HTML

Calculate PI based on the aggregate count of the points that lie within the circle

Demo : OpenMP Pi

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[LSU760000@n00 PA1]$ ./omcpi Enter the number of iterations used to estimate pi: 100000The current date/time is: Mon Mar 15 23:36:25 2010 thread 1 just did iteration 16665 the count is 3262 thread 5 just did iteration 66665 the count is 3263 thread 2 just did iteration 33332 the count is 6564 thread 6 just did iteration 83332 the count is 6596 thread 3 just did iteration 49999 the count is 9769 thread 7 just did iteration 99999 the count is 9810The current date/time is: Mon Mar 15 23:36:25 2010 the total count is 78534# of trials= 100000 , estimate of pi is 3.14136

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Creating Custom Communicators

• Communicators define groups and the access patterns among them

• Default communicator is MPI_COMM_WORLD• Some algorithms demand more sophisticated

control of communications to take advantage of reduction operators

• MPI permits creation of custom communicators

• MPI_Comm_create

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MPI Monte Carlo Pi Computation

Initialize MPIEnvironment

Receive Request

Compute Random Array

Send Array to Requestor

Last Request?

Finalize MPI

Y

N

Server

Initialize MPI Environment

WorkerMaster

Receive Error Bound

Send Request to Server

Receive Random Array

Perform Computations

Stop Condition Satisfied?

Finalize MPI

N

Y

Propagate Number of Points (Allreduce)

Initialize MPI Environment

Broadcast Error Bound

Send Request to Server

Receive Random Array

Perform Computations

Stop Condition Satisfied?

Print Statistics

N

Y

Propagate Number of Points (Allreduce)

Finalize MPI

Output Partial Result

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Monte Carlo : MPI - Pi (source code)#include <stdio.h>#include <math.h>#include "mpi.h“#define CHUNKSIZE 1000#define INT_MAX 1000000000#define REQUEST 1#define REPLY 2int main( int argc, char *argv[] ){ int iter; int in, out, i, iters, max, ix, iy, ranks[1], done, temp; double x, y, Pi, error, epsilon; int numprocs, myid, server, totalin, totalout, workerid; int rands[CHUNKSIZE], request; MPI_Comm world, workers; MPI_Group world_group, worker_group; MPI_Status status;

MPI_Init(&argc,&argv); world = MPI_COMM_WORLD; MPI_Comm_size(world,&numprocs); MPI_Comm_rank(world,&myid);

Initialize MPI environment

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Monte Carlo : MPI - Pi (source code) server = numprocs-1; /* last proc is server */ if (myid == 0) sscanf( argv[1], "%lf", &epsilon );

MPI_Bcast( &epsilon, 1, MPI_DOUBLE, 0, MPI_COMM_WORLD ); MPI_Comm_group( world, &world_group ); ranks[0] = server; MPI_Group_excl( world_group, 1, ranks, &worker_group );

MPI_Comm_create( world, worker_group, &workers ); MPI_Group_free(&worker_group);

if (myid == server) { do { MPI_Recv(&request, 1, MPI_INT, MPI_ANY_SOURCE, REQUEST, world, &status); if (request) {

for (i = 0; i < CHUNKSIZE; ) { rands[i] = random();

if (rands[i] <= INT_MAX) i++; }/* Send random number array*/MPI_Send(rands, CHUNKSIZE, MPI_INT, status.MPI_SOURCE, REPLY, world); }

}while( request>0 ); } else { /* Begin Worker Block */

request = 1; done = in = out = 0; max = INT_MAX; /* max int, for normalization */ MPI_Send( &request, 1, MPI_INT, server, REQUEST, world ); MPI_Comm_rank( workers, &workerid ); iter = 0;

Broadcast Error Bounds: epsilon

Create a custom communicator

Server process : 1. Receives request to generate a random ,2. Computes the random number array, 3. Send array to requestor

Worker process : Request the server to generate a random number array

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Monte Carlo : MPI - Pi (source code)while (!done) { iter++; request = 1; /* Recv. random array from server*/

MPI_Recv( rands, CHUNKSIZE, MPI_INT, server, REPLY, world, &status ); for (i=0; i<CHUNKSIZE-1; ) { x = (((double) rands[i++])/max) * 2 - 1;

y = (((double) rands[i++])/max) * 2 - 1;if (x*x + y*y < 1.0) in++;else out++;

} MPI_Allreduce(&in, &totalin, 1, MPI_INT, MPI_SUM, workers); MPI_Allreduce(&out, &totalout, 1, MPI_INT, MPI_SUM, workers); Pi = (4.0*totalin)/(totalin + totalout); error = fabs( Pi-3.141592653589793238462643); done = (error < epsilon || (totalin+totalout) > 1000000); request = (done) ? 0 : 1; if (myid == 0) { /* If “Master” : Print current value of PI */

printf( "\rpi = %23.20f", Pi );MPI_Send( &request, 1, MPI_INT, server, REQUEST, world );

} else { /* If “Worker” : Request new array if not finished */

if (request) MPI_Send(&request, 1, MPI_INT, server, REQUEST, world);

} }

MPI_Comm_free(&workers); }

Worker : Receive random number array from the Server

Worker: For each pair of x,y in the random number array, calculate the coordinates

Determine if the number is inside or out of the circle

Print current value of PI and request for more work

Compute the value of pi and Check if error is within threshhold

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Monte Carlo : MPI - Pi (source code)

if (myid == 0) { /* If “Master” : Print Results */

printf( "\npoints: %d\nin: %d, out: %d, <ret> to exit\n", totalin+totalout, totalin, totalout );getchar();

} MPI_Finalize();}

Print the final value of PI

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Demo : MPI Monte Carlo, Pi

[LSU760000@n00 PA1]$ mpiexec -np 4 ./monte 1e-7 pi = 3.14159265262020515053points: 12957000in: 10176404, out: 2780596