Page 90 - DCAP601_SIMULATION_AND_MODELING

Simulation and Modelling

Notes pseudo-random numbers, Monte Carlo methods are most suited to calculation by a computer.
Monte Carlo methods tend to be used when it is unfeasible or impossible to compute an exact
result with a deterministic algorithm.
Monte Carlo simulation methods are especially useful in studying systems with a large number
of coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and
cellular structures. More broadly, Monte Carlo methods are useful for phenomena with significant
uncertainty in inputs, such as the calculation of risk in business. These methods are also widely
used in mathematics: a classic use is for the evaluation of definite integrals, particularly
multidimensional integrals with complicated boundary conditions. It is a widely successful
method in risk analysis when compared to alternative methods or human intuition. When
Monte Carlo simulations have been applied in space exploration and oil exploration, actual
observations of failures, cost overruns and schedule overruns are routinely better predicted by
the simulations than by human intuition or alternative “soft” methods.

The term Monte Carlo method was coined in the 1940s by physicists working on nuclear weapon
projects in the Los Alamos National.

Overview

The Monte Carlo method can be illustrated as a game of battleship. First a player makes some
random shots. Next the player applies algorithms (i.e. a battleship is four dots in the vertical or
horizontal direction). Finally based on the outcome of the random sampling and the algorithm
the player can determine the likely locations of the other player’s ships.
There is no single Monte Carlo method; instead, the term describes a large and widely-used
class of approaches. However, these approaches tend to follow a particular pattern:

1. Define a domain of possible inputs.
2. Generate inputs randomly from the domain.
3. Perform a deterministic computation using the inputs.
4. Aggregate the results of the individual computations into the final result.

Example: The value of  can be approximated using a Monte Carlo method:
1. Draw a square on the ground, then inscribe a circle within it. From plane geometry, the
ratio of the area of an inscribed circle to that of the surrounding square is /4.
2. Uniformly scatter some objects of uniform size throughout the square. For example,
grains of rice or sand.
3. Since the two areas are in the ratio /4, the objects should fall in the areas in approximately
the same ratio. Thus, counting the number of objects in the circle and dividing by the total
number of objects in the square will yield an approximation for /4. Multiplying the
result by 4 will then yield an approximation for  itself.

Notice how the approximation follows the general pattern of Monte Carlo algorithms. First, we
define a domain of inputs: in this case, it’s the square which circumscribes our circle. Next, we
generate inputs randomly (scatter individual grains within the square), then perform a
computation on each input (test whether it falls within the circle). At the end, we aggregate the
results into our final result, the approximation of . Note, also, two other common properties
of Monte Carlo methods: the computation’s reliance on good random numbers, and its slow
convergence to a better approximation as more data points are sampled. If grains are purposefully
dropped into only, for example, the center of the circle, they will not be uniformly distributed,

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