Lecture 12 Monte Carlo methods in parallel computing

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Lecture 12 Monte Carlo methods in parallel
computing

• Parallel Computing
• Fall 2008

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What is Monte Carlo?

• 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
• 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

• A simple example to illustrate Monte Carlo
  principals
• Accelerate convergence using variance reduction
  techniques
• 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 lt N
• Pick two random numbers xi and yi
• If (xixi yiyi 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

• If message passing time is negligible
• 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 True
• Processors Size Time(sec) Deviation Error
• --------------------------------------------------
  ------
• 1 100,000 0.625 0.005 0.0018
• 2 50,000 0.25 0.005 0.0037
• 4 25,000 0.81 0.005 0.0061
• --------------------------------------------------
  ------

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Monte Carlo Estimates of Pi

• 1. Creating the random number generators
• for ( int i0 i lt num_dimension i) //
  create r.n. generator
• // the seeds are different for the processors
• seed (myRank 1) 100 i 10 1
• if ( seed 0 )
• randi new RandomStream(RANDOMIZE)
• else
• randi new RandomStream( seed )

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Monte Carlo Estimates of Pi

• 2. Each processors will run num_random / numProcs
  random walks
• for ( imyRank i lt num_random inumProcs )
  // N random walks
• 2a. Generating random numbers
• for ( int i10 i1 lt num_dimension i1)
• rani1 randi1- gt 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 ltlt "pi is " ltlt pi ltlt " sigma " ltlt 4
  sqrt( (pi/4 - (pi/4)(pi/4)) / num_random) ltlt "
  Error " ltlt abs(pi - PI25DT) ltlt endl

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End

• Thank you!