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Unformatted text preview: ty in nuclear reactors. In the 1940's, a formal foundation for the Monte Carlo method
was developed by von Neumann, who established the mathematical basis for probability
density functions (PDFs), inverse cumulative distribution functions (CDFs), and pseudorandom number generators. The work was done in collaboration with Stanislaw Ulam, who
realized the importance of the digital computer in the implementation of the approach. The
collaboration resulted from work on the Manhattan project, where the ENIAC was employed
in the calculation of yield Ulam et al., 1947 Ulam and Metropolis, 1949 Eckhard, 1987
Metropolis, 1987].
Individuals in the IBM corporation were pioneers in the eld of random number generation, perhaps because they were rst engaged in it due to their participation in the
Manhattan project, where Richard Feynman then directed their computing operations (a
fascinating exposition of their approach to performing largescale computing involving a
parallel approach exists in Richard Feynman's Surely You're Joking, Mr. Feynman). It
is interesting to note the extremely primitive computing environments in existence at that
time, and the challenges this presented to researchers (see Hurd, 1949).
Uses of Monte Carlo methods have been many and varied since that time. However, due
to computer limitations, the method has not yet fully lived up to its potential as discussed
by Metropolis Metropolis, 1985]. Indeed, this is re ected in the stages the method has
undergone in the elds of engineering. In the late 1950's and 1960's, the method was tested
in a variety of engineering elds Meuller, 1956 Ehrlich, 1959 Polgar and Howell, 1965
HajiSheikh, 1968 and Chandler et al., 1968]. At that time, even simple problems were
computebound. The method has since been extended to more complex problems Howell,
1968 Modest, 1978 Maltby, 1987a and 1987b, Maltby and Burns, 1988 and 1989 Crockett
et al., 1989]. Since then, attention was focused upon muchneeded convergence enhancement
procedures Kahn and Marshall, 1953 Emery and Carson, 1968 Burghart and Stevens, 1971
Lanore, 1971 Shamsunder et al., 1973 Zinsmeister and Sawyer, 1976 Larsen and Howell,
1986]. Many complex problems remained intractable through the seventies.
With the advent of highspeed supercomputers, the eld has received increased attention,
particularly with parallel algorithms which have much higher execution rates. In his Ph.D.
dissertation, Brown introduced the concept of the \event step" Brown, 1981], enabling
e cient vectorization of Monte Carlo algorithms where the particles do not interact. This
approach was later successfully exploited by several investigators. Martin et al. Martin et
al., 1986] reported speedups of a factor of ve on an IBM 3090 with vector units. Nearly linear
speedup was reported Sequent Computer Systems, 1985] on a parallel architecture for photon
tracing. Bobrowicz et al. Bobrowicz et al., 1984a and 1984b] obtained speedup factors of
from ve to eight in an algorithm where particles are accumulated in queues until e cient
vector lengths are obtained, allowing physics algorithms such as the Los Alamos benchmark
GAMTEB to be e ectively vectorized Burns et al., 1988]. Such advanced coding techniques Remarks 5 have enabled much bigger problems to be attacked, with improved accuracy. However, there
are still a host of problems which remain intractable, even with an e ectively vectorized
algorithm.
Moreover, some impediments to e ective vectorization have been identi ed and analyzed.
Zhong and Kalos Zhong and Kalos, 1983] analyzed the \straggler" problem, where few
particles persist for many event steps, inhibiting performance due to the overhead incurred
where vectors are \short." Pryor and Burns Pryor and Burns, 1988 Burns and Pryor, 1989]
reported, for one Monte Carlo problem where particles interact, speedups on the order of
half of those observed where the particles do not interact, albeit with a vectorized algorithm
of greatly increased complexity. The same problem has been attacked with di erent physics
McDonald and Bagano , 1988 Dagum, 1989] in a more e cient algorithm.
Good general references on Monte Carlo abound Beckenback, 1956 Hammersly and
Handscomb, 1964 Schreider, 1964 Kleinjnen, 1974 Rubenstein, 1981 Binder, 1984 HajiSheikh, 1988] in the literature. Most have a distinct bent  usually either statistics or physics.
Unfortunately, there is a dearth involving largescale engineering applications.
No tutorial on largescale Monte Carlo simulation can be complete without a discussion
of pseudorandom number generators. Where billions of random numbers are required, it
is essential that the generator be of long period, have good statistical properties and be
vectorizable. On 64bit machines, these criteria are usually satis ed with a multiplicative
generator if the constants are carefully chosen Kalos and Whitlock, 1986]. Indeed, we have
come a long way from the early days of random number generators Knuth, 1981]. For
example, Cray's FORTRAN callable generator ranf has a cycle length of 244, and is very
e ci...
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