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Lesson 1 Introduction to Monte Carlo
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Background and course overview
Monte Carlo methods are a branch of mathematics that involves the stochastic
solution of equations. In a sense, it is (and certainly feels like,
when you do it) an experimental approach to solving a problem.
When the analyst is trying to use a Monte Carlo approach to estimate
a physically measurable variable, the approach breaks itself down into
two steps:
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Devise a numerical experiment whose expected value would correspond to
the desired measurable value,
 .
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Run the problem to determine an estimate to this variable. We call
the estimate
 .
The first step can either be very simple or very complicated, based on
the actual physics of the situation. If the physical situation is
itself stochastic, the experimental design step is very simple:
Let the mathematical simulation mirror the physical situation. This
is called an analog simulation, since the calculation is a perfect analog
to the physical situation.
Lucky for us, the physical situation we are looking at -- the transport
of neutral particles -- is a stochastic situation. All we HAVE to
do to get a guess at a measurable effect from a transport situation is
to simulate EXACTLY the stochastic "decisions" that nature makes in particle
transport: the probabilities involved in particle birth, particle travel
through material media, particle death, and particle contribution to the
desired measurable value (also known as the "effect of interest").
For processes that are NOT inherently stochastic, the experimental design
is more complex and generally requires that the analyst:
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Derive an equation (e.g., heat transfer equation, Boltzmann transport equation)
from whose solution an estimate of the effect of interest can be inferred.
-
Develop a Monte Carlo method to solve the equation.
In this course we will do BOTH approaches. We will spend the first
1/3 of the course developing the mathematical tools that we will need,
then roughly 1/4 on the analog approach and 1/4 on the Boltzmann-based
approach. The final few classes will particularly look at the variance
reduction methods available in the MCNP computer program and will finish
with an analysis project that will serve as the final (take-home) exam.
Example: Finding 
Our first example will be a numerical estimation of  ,
based on use of a "dart board" approach. We know that the ratio of
the area of circle to the area of the square that (just barely) encloses
it
is going to be:
Knowing this, we can easily design an experiment that will deliver an
expected value of .
Let's set the origen at the center of the circle and the radius at 1; this
gives us an circle area of exactly .
The experiment will then be:
Choose a point at random inside the square by:
Score the result of a the trial: Consider a "hit" (score = 4) to
be the situation when the chosen point is inside the circle, i.e.,  ,
a "miss" scoring 0.
Note: This may seem a little strange to you, to have an experiment with
an "expected value" of ,
when the only possible real results are 0 and 4. We academics love
this sort of thing (i.e., expecting the impossible). By the way,
why is the score for a "hit" 4 instead of 1? Well, it is a little
like the three blind men and the elephant. A single trial of the
area of something inside a square of area 4 can only result in one of two
results: Either the inside object isn't there (area=0) or it fills
the box (area=4). These "scores" are nothing more than very crude
estimates of the circle's area.
3. Run the experiment a large number (N) of times,
with the final estimate of the circle's area being an average of the results:
 ,
where 
is the score for trial i.
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NE582 Monte Carlo