Transcript Lecture 1: Overview - City University of New York

Lecture 12 Monte Carlo
methods in parallel computing
Parallel Computing
Spring 2010
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What is Monte Carlo?
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Monte Carlo methods are a class of computational algorithms that
rely on repeated random sampling to compute their results.
Monte Carlo methods are often used in
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simulating physical and mathematical systems.
studying systems with a large number of coupled degrees of freedom,
such as fluids, disordered materials, strongly coupled solids, and
cellular structures (see cellular Potts model).
modeling phenomena with significant uncertainty in inputs, such as the
calculation of risk in business.
Mathematics: e.g. evaluation of definite integrals
Monte Carlo methods tend to be used when it is infeasible or
impossible to compute an exact result with a deterministic algorithm
(that is, intractable problems ).
Monte Carlo Algorithm is a randomized algorithm that may produce
incorrect results, but with bounded error probability.
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Applications of Monte Carlo
Radiation transport
Operations research
Nuclear criticality
Design of nuclear reactors
Design of nuclear weapons
Statistical physics
Phase transitions
Wetting and growth of thin films
Atomic wave functions and eigenvalues
Intranuclear cascade reactions Thermodynamic properties
Long chain coiling polymers
Reaction kinetics
Partial differential equations
Large sets of linear equations
Numerical integration
Uncertainty analysis
Development of statistical tests Cell population studies
Combinatorial problems
Search and optimization
Signal detection
WarGames
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Example 1 - Monte Carlo Integration
to Estimate Pi
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A simple example to illustrate Monte Carlo
principals
Accelerate convergence using variance
reduction techniques
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Use if expectation values
Importance sampling
Adapt to a parallel environment
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Monte Carlo Estimate of Pi
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Serial Monte Carlo Algoritm for Pi
Read N
Set SumG = 0.0
Do While i < N
Pick two random numbers xi and yi
If (xi*xi + yi*yi £ 1)
then SumG = SumG + 1
Endif
Enddo
Gbar = SumG / N
SigGbar = Sqrt(Gbar - Gbar2)
Print N, Gbar, SigGbar
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Parallelization of Monte Carlo
Integration
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If message passing time is negligible
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For same run time, error decreases by factor of
Sqrt(p)
For same accuracy, run time improves by a factor
of p
Should use same random number generator
on each node (easy for homogeneous
architecture)
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Monte Carlo Estimates of Pi
Number of Batch
CPU
Standard
Processors Size
Time(sec)
Deviation
-------------------------------------------------------1
100,000
0.625
0.005
2
50,000
0.25
0.005
4
25,000
0.81
0.005
--------------------------------------------------------
True
Error
0.0018
0.0037
0.0061
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Monte Carlo Estimates of Pi
1. Creating the random number generators
for ( int i=0; i < num_dimension; i++) { // create r.n. generator
// the seeds are different for the processors
seed = (myRank + 1) * 100 + i * 10 + 1;
if ( seed == 0 )
rand[i] = new RandomStream(RANDOMIZE);
else
rand[i] = new RandomStream( seed );
}
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Monte Carlo Estimates of Pi
2. Each processors will run num_random / numProcs random walks
for ( i=myRank; i < num_random; i+=numProcs ) { // N random walks
2a. Generating random numbers
for ( int i1=0; i1 < num_dimension; i1++) {
ran[i1] = rand[i1]- > next();
}
}
2b. Calculating for integration
if ( inBoundary( ran, num_dimension ) ) {
myPi += f( ran, num_dimension ) * pdf( ran, num_dimension );
count++;
}
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Monte Carlo Estimates of Pi
3. Calculating Pi from each processor
myPi = 4 * myPi / num_random;
MPI_Reduce(&myPi, &pi, 1, MPI_DOUBLE, MPI_SUM, 0, MPI_COMM_WORLD);
4. Output the results and standard deviation, sigma
if (myRank == 0) {
cout << "pi is = " << pi << " sigma = " << 4 * sqrt( (pi/4 - (pi/4)*(pi/4)) /
num_random) << " Error = " << abs(pi - PI25DT) << endl;
}
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End
Thank you!
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