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

Researchers should cite this work as follows:

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

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