STAN ULAM, JOHN VON NEUMANN,and the MONTE CARLO METHOD

by Roger Eckhardt

The Monte Carlo method is a sta-tistical sampling technique thatover the years has been appliedsuccessfully to a vast number of

scientific problems. Although the com-puter codes that implement Monte Carlohave grown ever more sophisticated, theessence of the method is captured in someunpublished remarks Stan made in 1983about solitaire.

“The first thoughts and attempts Imade to practice [the Monte Carlomethod] were suggested by a questionwhich occurred to me in 1946 as I wasconvalescing from an illness and play-ing solitaires. The question was whatare the chances that a Canfield solitairelaid out with 52 cards will come outsuccessfully? After spending a lot oftime trying to estimate them by pure

Los Alamos Science Special Issue 1987

combinatorial calculations, I wonderedwhether a more practical method than“abstract thinking” might not be tolay it out say one hundred times andsimply observe and count the numberof successful plays. This was alreadypossible to envisage with the begin-ning of the new era of fast computers,and I immediately thought of problemsof neutron diffusion and other ques-tions of mathematical physics, and moregenerally how to change processes de-scribed by certain differential equationsinto an equivalent form interpretableas a succession of random operations.Later... [in 1946, I ] described the ideato John von Neumann and we began toplan actual calculations.”

Von Neumann was intrigued.tical sampling was already well

Statis-known

in mathematics, but he was taken bythe idea of doing such sampling usingthe newly developed electronic comput-ing techniques. The approach seemed es-pecially suitable for exploring the behav-ior of neutron chain reactions in fissiondevices. In particular, neutron multiplica-tion rates could be estimated and used topredict the explosive behavior of the var-ious fission weapons then being designed.

In March of 1947, he wrote to Rob-ert Richtmyer, at that time the TheoreticalDivision Leader at Los Alamos (Fig. 1).He had concluded that “the statistical ap-proach is very well suited to a digitaltreatment,” and he outlined in some de-tail how this method could be used tosolve neutron diffusion and multiplica-tion problems in fission devices for thecase “of ‘inert’ criticality” (that is, ap-proximated as momentarily static config-

131

Fig. 1. The first and last pages of von Neumann’s remarkable letter to Robert Richtmyer are shown above, as well as a portion of his tentativecomputing sheet. The last illustrates how extensively von Neumann had applied himself to the details of a neutron-diffusion calculation.

132 Los Alamos Science Special Issue 1987

Monte Carlo

to me, therefore, that this approach willgradually lead to a completely satisfac-tory theory of efficiency, and ultimatelypermit prediction of the behavior of allpossible arrangements, the simple ones aswell as the sophisticated ones. ”

And so it has. At Los Alamos in 1947,the method was quickly brought to bearon problems pertaining to thermonuclearas well as fission devices, and, in 1948,Stan was able to report to the AtomicEnergy Commission about the applica-bility of the method for such things ascosmic ray showers and the study of theHamilton Jacobi partial differential equa-tion. Essentially all the ensuing work onMonte Carlo neutron-transport codes forweapons development and other applica-tions has been directed at implementingthe details of what von Neumann out-lined so presciently in his 1947 letter (see“Monte Carlo at Work”).

133

Monte Carlo

ated by making various decisions about els before interacting with a nucleus is unit interval (O, 1 ) into three subintervalsexponentially decreasing, making the se- in such a way that the probability of a

uniform random number being in a givensubinterval equals the probability of theoutcome assigned to that set.

In another 1947 letter, this time to StanUlam, von Neumann discussed two tech-niques for using uniform distributions ofrandom numbers to generate the desirednonuniform distributions g (Fig. 3). Thefirst technique, which had already been

134

May 21, 1947Mr. Stan UlamPost Office Box 1663Santa FeNew Mexico

Dear Stan :

Monte Carlo

very next free flight are accumulated aftereach collision. Random-Number

Generatorsby Tony Warnock

Random numbers have applications in many areas: simulation, game-playing,cryptography, statistical sampling, evaluation of multiple integrals, particle-transport calculations, and computations in statistical physics, to name a few.

Since each application involves slightly different criteria for judging the “worthiness”of the random numbers generated, a variety of generators have been developed, eachwith its own set of advantages and disadvantages.

Depending on the application, three types of number sequences might proveadequate as the “random numbers.” From a purist point of view, of course, a series ofnumbers generated by a truly random process is most desirable. This type of sequenceis called a random-number sequence, and one of the key problems is deciding whetheror not the generating process is, in fact, random. A more practical sequence is thepseudo-random sequence, a series of numbers generated by a deterministic processthat is intended merely to imitate a random sequence but which, of course, does notrigorously obey such things as the laws of large numbers (see page 69). Finally, aquasi-random sequence is a series of numbers that makes no pretense at being randombut that has important predefine statistical properties shared with random sequences.

Physical Random-Number Generators

Games of chance are the classic examples of random processes, and the firstinclination would be to use traditional gambling devices as random-number generators.Unfortunately, these devices are rather slow, especially since the typical computerapplication may require millions of numbers per second. Also, the numbers obtainedfrom such devices are not always truly random: cards may be imperfectly shuffled,dice may not be true, wheels may not be balanced, and so forth. However, in the early1950s the Rand Corporation constructed a million-digit table of random numbers usingan electrical “roulette wheel.” (The device had 32 slots, of which 12 were ignored; theothers were numbered from O to 9 twice.)

Classical gambling devices appear random only because of our ignorance of initialconditions; in principle, these devices follow deterministic Newtonian physics. Anotherpossibility for generating truly random numbers is to take advantage of the Heisenberguncertainty principle and quantum effects, say by counting decays of a radioactivesource or by tapping into electrical noise. Both of these methods have been used togenerate random numbers for computers, but both suffer the defects of slowness andill-defined distributions (however, on a different but better order of magnitude thangambling devices).

I 37

Monte Carlo

For instance, although each decay in a radioactive source may occur randomlyand independently of other decays, it is not necessarily true that successive counts inthe detector are independent of each other. The time it takes to reset the counter,for example, might depend on the previous count. Furthermore, the source itself

constantly changes in time as the number of remaining radioactive particles decreasesexponentially. Also, voltage drifts can introduce bias into the noise of electrical devices.

There are, of course, various tricks to overcome some of these disadvantages. Onecan partially compensate for the counter-reset problem by replacing the string of bitsthat represents a given count with a new number in which all of the original 1-1 and O-Opairs have been discarded and all of the original O-1 and 1-0 pairs have been changedto O and 1, respectively. This trick reduces the bias caused when the probability of aO is different from that of a 1 but does not completely eliminate nonindependence ofsuccessive counts.

A shortcoming of any physical generator is the lack of reproducibility. Repro-ducibility is needed for debugging codes that use the random numbers and for makingcorrelated or anti-correlated computations. Of course, if one wants random numbersfor a cryptographic one-time pad, reproducibility is the last attribute desired, and timecan be traded for security. A radioactive source used with the bias-removal techniquedescribed above is probably sufficient.

Arithmetical Pseudo-Random Generators

The most common method of generating pseudo-random numbers on the computeruses a recursive technique called the linear-congruential, or Lehmer, generator. Thesequence is defined on the set of integers by the recursion formula

or, in binary,

(1)

138

Monte Carlo

to form the normal number

1101110010111011110001001 101010111100110111101111 . . . .

If the number is blocked into 5-digit sets

11011,10010, 11101, 11100,01001, 10101,01111,00110,11 110, 11111,..., (2)

it becomes a sequence of numbers in base 2 that satisfy all linear statistical conditionsfor randomness. For example, the frequency of a specific 5-bit number is (1/2)5.

Sequences of this type do not “appear” random when examined; it is easy for aperson to guess the rule of formation. However, we can further disguise the sequenceby combining it with the linear-congruence sequence generated earlier (Seq. 1). We dothis by performing an exclusive-or (XOR) operation on the two sequences:

Los Alamos Science Special Issue 1987 1 3 9

Monte Carlo

and

11011, 10010,11101,11100,01001, 10101,01111,00110, 11110,111 11,... (2)

yield

Of course, if Seq. 3 is carried out to many places, a pattern in it will also becomeapparent. To eliminate the new pattern, the sequence can be XOR’ed with a thirdpseudo-random sequence of another type, and so on.

This type of hybrid sequence is easy to generate on a binary computer. Althoughfor most computations one does not have to go to such pains, the technique is especially

attractive for constructing “canonical” generators of apparently random numbers.A key idea here is to take the notion of randomness to mean simply that the

sequence can pass a given set of statistical tests. In a sequence based on normalnumbers, each term will depend nonlinearly on the previous terms. As a result, thereare nonlinear statistical tests that can show the sequence not to be random. In particular,a test based on the transformations used to construct the sequence itself will fail. But,the sequence will pass all linear statistical tests, and, on that level, it can be consideredto be random.

What types of linear statistical tests are applied to pseudo-random numbers?Traditionally, sequences are tested for uniformity of distribution of single elements,pairs, triples, and so forth. Other tests may be performed depending on the type ofproblem for which the sequence will be used. For example, just as the correlationbetween two sequences can be tested, the auto-correlation of a single sequence can betested after displacing the original sequence by various amounts. Or the number ofdifferent types of “runs” can be checked against the known statistics for runs. Anincreasing run, for example, consists of a sequential string of increasing numbersfrom the generator (such as, 0.08, 0.21, 0.55, 0.58, 0.73, . . .). The waiting timesfor various events (such as the generation of a number in each of the five intervals(o, 0,2), (0.2, 0,4), ..., (0.8, 1)) may be tallied and, again, checked against the knownstatistics for random-number sequences.

If a generator of pseudo-random numbers passes these tests, it is deemed to be a“good” generator, otherwise it is “bad.” Calling these criteria “tests of randomness” ismisleading because one is testing a hypothesis known to be false. The usefulness ofthe tests lies in their similarity to the problems that need to be solved using the streamof pseudo-random numbers. If the generator fails one of the simple tests, it will surelynot perform reliably for the real problem. (Passing all such tests may not, however, beenough to make a generator work for a given problem, but it makes the programmerssetting up the generator feel better.)

140 pecial Issue 1987

Monte Carlo

the basis of the Metropolistechnique of evaluating integrals by the Monte Carlo method,

Now if the points are taken from a random or a psuedo-random sequence, thestatistical uncertainty will be proportional to 1/@. However, if a quasi-random se-quence is used, the points will occupy the coordinate space with the correct distributionbut in a more uniform manner, and the statistical uncertainty will be proportional tol/N. In other words, the uncertainty will decrease much faster with a quasi-randomsequence than with a random or pseudo-random sequence.

How are quasi-random sequences generated? One type of sequence with a veryuniform distribution is based on the radical-inverse function. The radical-inverse

uniform distribution and is useful in mutiple integration or multi-dimensional sampling.There are many other types of random, pseudo-random, or quasi-random sequences

than the ones I have discussed here, and there is much research aimed at generatingsequences with the properties appropriate to the desired application. However, theexamples I have discussed should illustrate both the approaches being taken and theobstacles that must be overcome in the quest of suitable “random” numbers. ■

141

Monte Carlo

Monte Carloat Work

by Gary D. Doolenand John Hendricks

Every second nearly 10,000,000,000 “random” numbers are being generated oncomputers around the world for Monte Carlo solutions to problems that StanUlam first dreamed of solving forty years ago. A major industry now exists

that has spawned hundreds of full-time careers invested in the fine art of generatingMonte Carlo solutions—a livelihood that often consists of extracting an answer out of anoisy background. Here we focus on two of the extensively used Monte Carlo solvers:MCNP, an internationally used neutron and photon transport code maintained at LosAlamos; and the “Metropolis” method, a popular and efficient procedure for computingequilibrium properties of solids, liquids, gases, and plasmas.

MCNP

In the fifties, shortly after the work on the Monte Carlo method by Ulam, vonNeumann, Fermi, Metropolis, Richtmyer, and others, a series of Monte Carlo transportcodes began emerging from Los Alamos. The concepts on which these codes werebased were those outlined by von Neumann (see “Stan Ulam, John von Neumann, andthe Monte Carlo Method”), but a great deal of detailed work was needed to incorporatethe appropriate physics and to develop shorter routes to statistically valid solutions.

From the beginning the neutron transport codes used a general treatment of the ge-ometry, but successive versions added such features as cross-section libraries, variance-reduction techniques (essentially clever ways to bias the random numbers so that theguesses will cluster around the correct solution), and a free-gas model treating ther-malization of the energetic fission neutrons. Also, various photon transport codes weredeveloped that dealt with photon energies from as low as 1 kilo-electron-volt to thehigh energies of gamma rays. Then, in 1973, the neutron transport and the photontransport codes were merged into one. In 1977 the first version of MCNP appearedin which photon cross sections were added to account for production of gamma raysby neutron interactions. Since then the code has been distributed to over two hundredinstitutions worldwide.*

The Monte Carlo techniques and data now in the MCNP code represent over threehundred person-years of effort and have been used to calculate many tens of thousandsof practical problems by scientists throughout the world. The types of problems includethe design of nuclear reactors and nuclear safeguard systems, criticality analyses, oilwell logging, health-physics problems, determinations of radiological doses, spacecraftradiation modeling, and radiation damage studies. Research on magnetic fusion hasused MCNP heavily.

The MCNP code features a general three-dimensional geometry, continuous energyor multigroup physics packages, and sophisticated variance reduction techniques. Evenvery complex geometry and particle transport can be modeled almost exactly. In fact,the complexity of the geometry that can be represented is limited only by the dedicationof the user.

*The MCNP code and manual can be obtained from the Radiation Shielding Information Center (RSIC),P.O. Box X, Oak Ridge, TN 37831.

142 Los Alamos Science Special Issue 1987

Monte Carlo

PAIR-DISTRIBUTION FUNCTION

This plot gives the probability of pairs of

charged particles in a plasma being sep-arated by a certain distance. The prob-

abilities are plotted as a function of the

distance between the pair of particles (in-creasing from left to right) and tempera-ture (decreasing from front to back). Atthe left edge, both the distance and theprobability are zero; at the right edge, theprobability has become constant in value.Red indicates probabilities below this con-stant value, yellow and green above. Asthe temperature of the plasma decreases,lattice-like peaks begin to form in the pair-

distribution function. The probabilities, gen-erated with the Metropolis method describedin the text, have been used for precise tests

of many theoretical approximations for plas-ma models.

Los Alamos Science Special Issue 1987 143