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Illinois PHYS 466, Lecture 10: Sampling

By David M. Ceperley1, Omar N Sobh1

1. University of Illinois at Urbana-Champaign

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Abstract

Fundamentals of Monte Carlo

What is Monte Carlo?

  • Named at Los Alamos in 1940’s after the casino.
  • Any method which uses (pseudo)random numbers> as an essential part of the algorithm.
  • Stochastic - not deterministic!

  • A method for doing highly dimensional integrals by sampling the integrand.
  • Often a Markov chain, called Metropolis MC.

Content:

  • Simple example: Buffon’s needle - Monte Carlo determination of π
  • MC is advantageous for high dimensional integrals -the best general method
  • Improved Numerical Integration
  • Other reasons to do Monte Carlo
  • Probability Distributions
  • Mappings of random variables
  • What is Mapping Doing?
  • Interpreting the Mapping
  • Example: Drawing from Normal Gaussian
  • Reminder: Gauss’ Central Limit Theorem
  • Cumulants: κn Mean = κ1 Variance= κ2 Skewness = κ3 Kurtosis= κ4
  • Approach to normality
  • Conditions on Central Limit Theorem
  • 2d histogram of occurrences of means

Credits

These lecture were breezed and uploaded by Omar Sobh

Cite this work

Researchers should cite this work as follows:

  • David M. Ceperley; Omar N Sobh (2009), "Illinois PHYS 466, Lecture 10: Sampling," http://nanohub.org/resources/6507.

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