Monte Carlo Simulation Part.2 Metropolis Algorithm

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Monte Carlo SimulationPart.2 Metropolis Algorithm

• Dept. Phys. Tunghai Univ.
• Numerical Methods
• C. T. Shih

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Simple/Importance Sampling

• ???? MC ??????,????????? f(x)constant,????????,??
  ???????
• ???,?? f(x) ? delta ??,????????,????????
• ????,????????,f(x) ??????,? MC ????,??f(x)??????,?
  MC ???????

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Weight Function w(x)

• ??,? f(x) ?????,?? MC ???,???? f(x) ??????????
• ??f(x) ??,??f(x)dx?????,???????????????
• ?????? x ??????,? f(x) ??????????
• Weight distribution function w(x),????x??????

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Weight Function w(x)

• w(x) ?????,????????w(x)dx1
• ?? x ????? w(x) ??,????????????????????????M?x,??
  ????

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Metropolis ???

• ???????? x,?? x ????? w(x)?????
• ?????????? x,altxltb
• ????????? x,altxltb
• ????????? P(x?x) w(x)/w(x)
• ? P ? 1,??? x ??? x ?
• ? P lt 1,?????? y,? yltP,??? x ???
  x,?????????????? x
• ?????????? x,????? M ?,??????? x ???? f(x)/w(x)
  ???,??????????

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f(x) ? w(x)

• ??? w(x) ???? f(x) ??,???? w(x) ????
  ?w(x)dx1,????? w(x) ??? ? ???????
• ???? f(x) ????? probability distribution
  function(??????),??????????????????
• ??????? Metropolis algorithm ????? f(x) ???? M ? x