Page 91 - DCAP601_SIMULATION_AND_MODELING

Unit 6: Discrete System Simulation (III)

and so our approximation will be poor. An approximation will also be poor if only a few grains Notes
are randomly dropped into the whole square. Thus, the approximation of ð will become more
accurate both as the grains are dropped more uniformly and as more are dropped.
History

The name “Monte Carlo” was popularized by physics researchers Stanislaw Ulam, Enrich Fermi,
John von Neumann, and Nicholas Metropolis, among others; the name is a reference to the
Monte Carlo Casino in Monaco where Ulam’s uncle would borrow money to gamble. The use of
randomness and the repetitive nature of the process are analogous to the activities conducted at
a casino.
Random methods of computation and experimentation (generally considered forms of stochastic
simulation) can be arguably traced back to the earliest pioneers of probability theory (see, e.g.,
Buffon’s needle, and the work on small samples by William Gosset), but are more specifically
traced to the pre-electronic computing era. The general difference usually described about a
Monte Carlo form of simulation is that it systematically “inverts” the typical mode of simulation,
treating deterministic problems by first finding a probabilistic analog. Previous methods of
simulation and statistical sampling generally did the opposite: using simulation to test a
previously understood deterministic problem. Though examples of an “inverted” approach do
exist historically, they were not considered a general method until the popularity of the Monte
Carlo method spread.

Perhaps the most famous early use was by Enrico Fermi in 1930, when he used a random method
to calculate the properties of the newly-discovered neutron. In the 1950s they were used at Los
Alamos for early work relating to the development of the hydrogen bomb, and became
popularized in the fields of physics, physical chemistry, and operations research. The Rand
Corporation and the U.S. Air Force were two of the major organizations responsible for funding
and disseminating information on Monte Carlo methods during this time, and they began to
find a wide application in many different fields.

Uses of Monte Carlo methods require large amounts of random numbers, and it was their use
that spurred the development of pseudorandom number generators, which were far quicker to
use than the tables of random numbers which had been previously used for statistical sampling.

Notes Monte Carlo methods were central to the simulations required for the Manhattan
Project, though were severely limited by the computational tools at the time. Therefore, it
was only after electronic computers were first built (from 1945 on) that Monte Carlo
methods began to be studied in depth.

Applications

As mentioned, Monte Carlo simulation methods are especially useful for phenomena with
significant uncertainty in inputs and in studying systems with a large number of coupled degrees
of freedom. Specific areas of application include:

Physical Sciences

Monte Carlo methods are very important in computational physics, physical chemistry, and
related applied fields, and have diverse applications from complicated quantum chromo dynamics
calculations to designing heat shields and aerodynamic forms. The Monte Carlo method is
widely used in statistical physics, in particular, Monte Carlo molecular as an alternative for

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