Monte Carlo

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Monte Carlo

• A Monte Carlo method can be loosely described as
  a statistical method used in simulation of data.
• And a simulation is defined to be a method that
  utilizes sequences of random numbers as data.

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Monte Carlo Method

• Doesnt require sophisticated mathematical model
  (no odes or pdes) just some probability density
  functions.
• The Monte Carlo method usually adhere to natural
  phenomenon and thus doesnt require a complex
  mathematical model.
• Example..

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Monte Carlo Algorithm

• Probability distribution function the
  physical/mathematical system described by a set
  of pdfs.
• Random number generator a source of random
  numbers with uniform distribution on the unti
  interval.
• Sampling rule a prescription for sampling from
  the specified pdfs.

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Monte Carlo Algorithm, cont.

• Scoring/Tallying outcomes must accumulate into
  overall tallies or scores for quantities of
  interest.
• Error estimation measure of the statistical
  error (variance) as a function of the number of
  trials.

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Monte Carlo Algorithm III

• Variance Reduction techniques methods for
  reducing the variance in the estimated solution
  to reduce computational time for a Monte Carlo
  method.
• Parallelization/Vectorization algorithms
  designed to use Monte Carlo methods on advanced
  computing platforms.

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Monte Carlo Example Estimating Pi

• This example will show how a Monte Carlo method
  can be used to approximate the value of pi.
• Step 1 assume a unit circle enclosed by a unit
  square.

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Pi, cont.

• Step 2 - Look at the first quadrant.
• Step 3 start throwing darts at the picture,
  count the number of hits inside the circular arc
  versus the total hits inside the square.

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More Pi

• Step 4 after a little geometry review, the area
  of the square is rr and the area of the circle
  is 1/4pirr.
• Step 5 a little algebraic manipulation and you
  have that pi 4(darts shaded/darts total).