Table Of ContentN. P. BUSLENKO, D. I. GOLENKO, Yu. A. SHREIDER,
I. M. SOBOL' AND V. G. SRAGOVICH
THE
MONTE CARLO
METHOD
The Method of Statistical Trials
EDITED BY
Yu. A. SHREIDER
TRANSLATED FROM THE RUSSIAN
BY
G.J.TEE
TRANSLATION EDITED BY
D.M.PARKYN
PERGAMON PRESS
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Copyright © 1966
Pergamon Press Ltd.
First English edition 1966
Translated from MeTOfl CTaTiiCTiraecKHX HcnwTaiiHft —
(Μβτο^ MOHTe-Kapjio), No. 5 in the series Spravochnaya
Matematicheskaya Biblioteka, published by Fizmatgiz,
Moscow, 1962
Library of Congress Catalog Card No. 64-8052
2156/66
PREFACE TO THE ENGLISH EDITION
THE advent of modern electronic computers has led to the develop-
ment of new techniques of investigation, based on new principles.
By means of experiments with computers, models of complex
physical systems can rapidly be studied in a manner which is
relatively simple and inexpensive. A most important feature is the
possibility of simulating random processes on a computer. The so-
called method of statistical trials (or Monte Carlo method) is, in
essence, a system of techniques which enables us to model such
processes conveniently in a machine. This method can be used
effectively for various degrees of simulation, ranging from cases in
which actual physical systems are simulated (describing them in
some language which is specific for the particular problem given)
to investigations of classical mathematical problems (such as
systems of linear algebraic equations).
It is interesting to note that computers have led to a novel
revolution in mathematics. Whereas previously an investigation of
a random process was regarded as being complete as soon as it was
reduced to an analytic description, nowadays it is convenient in
many cases to solve an analytic problem by reducing it to a corre-
sponding random process and then simulating that process.
In this present book, the authors have set themselves the aim
of describing systematically the fundamental concepts and tech-
niques of the Monte Carlo method, together with its range of appli-
cation. The authors are grateful to Pergamon Press and to the
translator Mr. G. J. Tee, who kindly accepted all proposals for
improving this book. A number of additions made by the authors
have been incorporated into the translation of this book. In
Chapter I, § 4 and § 5 have been enlarged considerably, whilst the
new § 7 and § 8 are translated from Chapters VI and IX of another
book [11] written by the same author. Also, Chapter VII contains a
new § 7, written especially by V. G. Sragovich for this English edition.
The authors will be most gratified if this book proves to be useful
for English readers.
Yu. A. SHREIDER
vii
PREFACE
THE present volume of the series "Library of Mathematical Hand-
books" f is devoted to the so-called method of statistical trials (or
Monte Carlo Method). In contrast to the volumes published previ-
ously, which were devoted to the classical branches of mathematics
with a strictly delimited range of problems, an established termi-
nology and a tradition of exposition, the mathematical methods
considered in this volume have been developed only during the
past 13 years.+
These methods, which are applicable to the most widely diverse
branches of computational mathematics, are linked by an idea
which is common to them all.
Their basis lies in simulating statistical experiments by means
of computational techniques, and recording numerical characteris-
tics obtained from these experiments. Hence all of these methods
have the general title of method of statistical trials, or Monte Carlo
method. The solution of numerical problems by this method is
closer in spirit to physical experimentation than to the classical
numerical methods.
Indeed, the error of the Monte Carlo method cannot be estimated
sufficiently well a priori, and as a rule it is found by means of
investigating the standard deviation of the quantities being simu-
lated. The solution might not be reproducible in detail. The solu-
tion is stable with respect to isolated errors in the operation of the
computer which is employed.
The purpose of this book is to expound the basic features of
the method, to give an adequate account of the techniques used in
the Monte Carlo method and of typical examples, and to illustrate
the principal fields of application of the Monte Carlo method.
This objective explains the structure of the book.
In Chapter I (written by Yu. A. Shreider) the basic features of
f Publisher's Note: Tis is a reference to the series of Russian mathe-
matics books.
Φ Essentially, we may speak of the application of the Monte Carlo method
only after the publication of the work of N. Metropolis and S. Ulam in 1949
[203], The term "Monte Carlo" first appeared in that work.
ix
X PREFACE
the method are explained and typical examples are considered of
its application to simple problems in computational mathematics.
Chapter II (written by I. M. Sobol') contains a very detailed
study of the computation of multi-dimensional integrals, which are
(in a certain sense) a fundamental field of application of the method.
A variety of examples of statistical modelling of integrals is ana-
lysed therein, and the accuracy of the computations is investigated.
Chapter III (written by I. M. SoboF) and Chapter V (written by
V. G. Sragovich) are devoted to the applications of the Monte Carlo
method in those branches of physics and technology where it has
been widely and successfully used. In fact, the Monte Carlo method
was devised for solving problems in neutron physics. This is because
the random processes which are investigated by the method (the
transmission of beams of neutrons through scattering media, fluc-
tuation of radio signals, etc.) are so complicated that their analytical
solution is practically impossible. But the statistical simulation
of such processes on a digital computer enables successful investi-
gations to be made. These processes have specific peculiarities
and many special devices were employed for modelling them, so that
it proved to be convenient to divide them into two separate chapters.
Chapter IV (written by N. P. Buslenko) is devoted to the appli-
cation of the Monte Carlo method to the investigation of servicing
processes. This branch, which has been developed during the last
few years, is concerned with the simulation of complex systems of
control and operations research, and has great prospects in mathe-
matical cybernetics.
In Chapters VI (written by D. I. Golenko) and VII (§§ 1, 2, 3 and
5 written by N. P. Buslenko, § 6 written by V. G. Sragovich, and
§ 4 written jointly by N. P. Buslenko and V. G. Sragovich)
methods are considered for organizing statistical experiments on
universal digital computers. The questions considered in these
chapters are essential for any application of the method of statisti-
cal trials.
Two illuminating examples yet remain of fields of application of
the Monte Carlo method: the investigation of learning machines
and the study of the reliability of complicated electronic apparatus
(e.g. computers). However, these and some other applications are
still in the early stages of development, so that it would be difficult
to include the corresponding material in a handbook. Questions
relating to the first of them were partially clarified in the book
PREFACE xi
Stochastic Models of Learning, by R. Bush and F. Mosteller, a
Russian translation of which was published by Fizmatgiz in 1962.
In view of the fact that the bibliographies for the various chap-
ters overlap considerably, and also for convenience in handling,
a general combined bibliography of the Monte Carlo method (as-
sembled by I. M. Sobol') is given at the end of the book. All refer-
ences in the text refer to this bibliography. Literature of a more
general character, which is not concerned directly with the Monte
Carlo method, is given after this bibliography, sharing a common
numeration with it.f
General questions concerning the Monte Carlo method were dis-
cussed in a recently published book by N. P. Buslenko and Yu. A.
Shreider [11]. The present volume differs from it giving a more
detailed treatment of the material, and by the display of a wider
range of applications.
We do not consider in this handbook any questions concerning
the peculiarities of the structure of electronic digital computers
which may be employed for the solution of problems by the Monte
Carlo method. In particular, we do not consider questions concern-
ing the construction of specialized computers.
A large group of authors collaborated in the writing of this hand-
book. Notwithstanding the extensive work which was done to
harmonize the separate chapters, there remain a few inevitable
discrepancies in the contents and in the style of presentation.
This handbook is designed for a wide circle of readers, ranging
from those who are interested in the fundamental applications of
the method, to those who are concerned with comparatively limit-
ed problems of the peculiarities of simulating physical processes.
If we do not count Chapters III and V, which are intended for
readers who are familiar with certain general facts concerning neu-
tron physics and radio technology, then an understanding of the
major part of the material expounded in this book requires from
the reader a mathematical basis equivalent to a college course of
mathematical analysis, and also a knowledge of the elements of
the theory of probability. It is assumed that the reader is familiar
with the basic facts concerning random events and quantities and
their stochastic characteristics (the probability of the occurrence
f No such bibliography of more general literature appears in the Russian
original. [G. J. T.]
xii PREFACE
of an event, mathematical expectation, variance). Moreover, it is
necessary to have some knowledge of the law of normal distribu-
tion, Lyapunov's theorem and, for certain chapters, of the ele-
ments of mathematical statistics. It is also most desirable to be
familiar with Markov processes.
The authors trust that this handbook will prove to be useful to
those people who are interested in the Monte Carlo method and its
applications; and, what is no less important, that it will introduce
many readers to the idea of the usefulness of applying the Monte
Carlo method for solving their problems. The authors express their
gratitude to V. D. Rozenknop, for having given much valuable
advice concerning the form of the text.
Yu. A. SHBEIDER
CHAPTER I
PRINCIPLES OF THE
MONTE CARLO METHOD
§ 1. Definition, and Simple Examples of the Application, of the
Monte Carlo Method
The Monte Carlo method (or method of statistical trials) consists
of solving various problems of computational mathematics by
means of the construction of some random process for each such
problem, with the parameters of the process equal to the required
quantities of the problem. These quantities are then determined
approximately by means of observations of the random process
and the computation of its statistical characteristics, which are
approximately equal to the required parameters.
For example, the required quantity x might be the mathematical
expectation M ξ of a certain random variable. The Monte Carlo
method for determining the approximate value of the quantity x
consists of an i\f-fold sampling of the value of the variable ξ in a,
series of independent tests: ξ , ξ , ..., £jy, and the computation of
χ 2
their mean value:
. li +l + - +ftv
= 2
* N
According to the law of large numbers,
I *, M£ = x,
with a probability which is as close as required to unity for suf-
ficiently large N. Accordingly the quantity |, which has been deter-
mined by observation of the random process, is approximately
equal to the required quantity x.
We shall present a simple example.
Let the probability w be computed that the number of hits
in ten throws at an " apple -range "f be even.
t of. "Coconut shy". [G. J. T.]
1 MCM 1
2 PRINCIPLES OF THE MONTE CARLO METHOD [I, 1]
If the probability of hitting an apple with one throw is p, then the
required probability w may be computed by the formula:
w = ZCf p2k{\ -p)10-*K (1.1)
0
Formula (1.1) is obvious if we observe that the general term,
operated upon by the summation sign, is the probability that the
number of hits is exactly 2 k.
If we use published tables of the combinations C\Q, then the
computation of the probability by formula (1.1) requires 20 multi-
plications and 6 additions.
Instead of this, we could have performed N series of 10 throws
each and then have formed the number L as the sum of the num-
bers of cases in which there were an even number of hits in a se-
quence of ten throws. With N sufficiently large, the fraction LjN
will be a good representation of the required probability w. It will
be shown later that in order to obtain w reliably to two decimal
places, it would be necessary to perform about 10,000 series of
tests with 10 throws each.
In this case it would be simpler to perform the simple computa-
tion of formula (1.1), rather than perform 100,000 throws.
This example is simply the determination of an unknown quan-
tity by actual experiment. In the more strict sense of the term, the
Monte Carlo method is defined as the construction of an artificial
random process possessing all the necessary properties, but which is
in principle realizable by means of ordinary computational appa-
ratus: pencil, paper, tables, computers and sometimes simple
apparatus for generating random numbers (the so-called random
number devices; cf. Ch. VI).
In practice the Monte Carlo method has found wide application
only on powerful computing machines. As we shall see later, the
properties of this method make it peculiarly suitable for realization
on digital computers. Usually, universal digital computers are
used for performing calculations by the Monte Carlo method, but
in a number of cases it is worthwhile to construct specialized
machines for special classes of problems.
We shall now describe a solution of the above problem of com-
puting the probability w, by means of simple computational appa-
ratus. Instead of throwing shots, we shall use a top and a clock
[I, 1] DEFINITION AND SIMPLE EXAMPLES 3
with a seconds hand, together with pencil and paper. For simplicity
we shall assume that p = 1/5. Instead of throwing ten shots we
shall spin the top ten times. We shall observe the clock at the in-
stant when the top tips onto the floor, and note the position of the
seconds hand. If the seconds hand reads a value τ in the range
0 < τ < 12, then we shall score a "hit". If in the course of a series
of ten spins we score an even number of hits, then we shall register
"success". We shall perform N series of such tests, each consisting
of ten spins of the top. Let L of them be "successful". Then the
random variable LjN will be distributed in exactly the same man-
ner as the analogous variable described above in the case of actual
throws, and accordingly it will be approximately equal to the
required probability w.
Although this experiment of spinning a top requires less labour
than throwing the shots it is not, however, more rapid in perfor-
mance than the actual throws. It would be necessary to use an
electronic computer for a speedy solution.
In order to determine the characteristics of a random process,
it is modelled with the aid of a computer and its statistical charac-
teristics are computed, then printed by the machine as the solution
of the problem.
In particular, the example chosen by us could be realized in a
computer in the following manner. Many machines have random
number devices (cf. Ch. VI), enabling us at each instant to find
the value ξ of a random variable which is uniformly distributed in
the range [0, 1]. Instead of throwing at a target or spinning a top,
we shall take a value of ξ from the device and test whether it
satisfies the inequaUty ξ < p. If it is fulfilled then we shall regard
it as constituting a "hit". It is not difficult to see that the proba-
bility of the fulfilment of the inequaUty ξ < p (which is the
probabiUty of a "hit") is p.
We take a series of ten numbers ξ, and if an even number of
"hits" occurs in this series we shall register "success". Let L
series out of N be successful. We may say exactly the same con-
cerning the variable LjN as was said earUer. The imitation of one
throw in the above manner requires two instructions on the com-
puter "Strela".
Thus the imitation of the entire process with N = 10,000 series
to be accompUshed would require of the order of 200,000 operations,
i.e. 100 seconds working time on "Strela". This is already much
1*
