Transcript Monte Carlo Simulation - Washington University in St. Louis

MONTE CARLO SIMULATION
Presented by: Zhenhuan Sui
Introduction From A Simple Application
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Facts
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It is the computational algorithm that rely on repeated
random sampling to compute its result
Investigating radiation shielding and the distance that
neutrons would likely travel through various materials at
Los Alamos Scientific Laboratory in Manhattan Project
John von Neumann and Stanislaw Ulam: modeling the
experiment on a computer using chance
The name is a reference to the Monte Carlo Casino in
Monaco where Ulam's uncle would borrow money to
gamble
1950s’ hydrogen bomb
General Steps For MCS
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Define a domain of possible inputs
Generate inputs randomly from the domain
using a certain specified probability distribution
Perform a deterministic computation using the
inputs
Aggregate the results of the individual
computations into the final result
suited to calculation by a computer
statistical simulation method
http://en.wikipedia.org/wiki/Monte_Carlo_method#Monte_Carlo_and_random_numbers
Elements
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probability density function
random number generator
sampling rules
simulation results
uncertainties estimation
technology to reduce the variance
Buffon's Needle
The uniform probability density
function of x between 0 and t /2 is
2/t dx
The uniform probability density
function of θ between 0 and π/2 is
2/π d θ
http://upload.wikimedia.org/wikipedia/comm
ons/f/f6/Buffon_needle.gif
http://en.wikipedia.org/wiki/Buffon%27s_needle
Further For π
http://en.wikipedia.org/wiki/Buffon%27s_needle
Steps For Applications
• based on the properties of the systems we
want to solve, we construct the theoretical
models which can describe the properties.
• try to get the probability density functions of
some properties for the models.
• From the probability density functions, we can
produce some random samplings and get
some simulation results.
• analyze the results and do predictions for
some properties of the systems.
Applications
Physics:
• quantum chromodynamics
• statistical physics
• particle physics
Mathematics:
• Monte Carlo integration: algorithms for the approximate
evaluation of definite integrals, multidimensional ones
• numerical optimization
Finance and business:
• calculate the value of companies
• evaluate investments in projects
• evaluate financial derivatives
• risk management, there would be a lot of variables in the
equations
• Quasi-Monte Carlo methods in finance
RESOURCES
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http://en.wikipedia.org/wiki/Monte_Carlo_method#Finance_and_business
http://en.wikipedia.org/wiki/Monte_Carlo_method
http://en.wikipedia.org/wiki/Quasi-Monte_Carlo_methods_in_finance
http://baike.baidu.com/view/7775.html
http://www.virtualphantoms.org/egs4/temp/mchome.htm
http://www.linuxsir.org/bbs/thread288992.html