To the Beginning of the Third Millennium NOTES ON PHILOSOPHY OF THE MONTE CARLO METHOD I. Elishakoff UDC 539.3 Some personal (and inevitably biased) thoughts pertinent to the Monte Carlo method are shared with the reader; in this mass production process of papers we are pausing and asking ourselves some nagging questions. Various aspects of the philosophy of the method are discussed. Its role in stochastic mechanics is elucidated and some questions are posed in view of promoting a further constructive discussion. Keywords: Monte Carlo method, some nagging questions, various aspects of philosophy of the method, stochastic mechanics 1. Introduction. The name “Monte Carlo method” dates from about 1944 and is due to von Neumann and Ulam, who introduced it as a code name for their work on neutron diffusion problems at the Los Alamos Scientific Laboratory. The name itself was selected because roulette (with which the casino town Monte Carlo is traditionally associated) is one of the simplest tools for generating random numbers. Systematic development dates from 1949, with publication of the paper by Metropolis and Ulam. McCracken [54] writes: “During World War II physicists at the Los Alamos Scientific Laboratory came to a knotty problem of the behavior of neutrons. How far would neutron travel through various materials? The question had a vital bearing on shielding and other practical considerations. But it was extremely complicated to answer. To explore it by experimental trial and error would have been expensive, time-consuming and hazardous. On the other hand, the problem seemed beyond the reach of theoretical calculations. The physicists had most of the necessary basic data: They knew the average distance a neutron of a given speed would travel in a given substance before it collided with an atomic nucleus, what the probabilities were that the neutron would bounce off instead of being absorbed by the nucleus, how much energy the neutron was likely to lose after a collision, and so on. However, to sum all of this up in a practicable formula for predicting the outcome of a whole sequence of such events was impossible. At this crisis, the mathematicians, John van Neumann and Stanlas Ulam, cut the Gordian knot with a remarkably simple stroke. They suggested a solution that in effect amounts to submitting the problem to a roulette wheel. Step by step, the probabilities of the separate events merged into a composite picture which gives an approximate but a workable answer to the problem.” Various definitions of the Monte Carlo method have been given. Halton [29] defines it as follows: “The Monte Carlo method is defined as representing the solution of a problem as a parameter of a hypothetical population, and using a random sequence of numbers to construct a sample of the population from which statistical estimates of the parameters can be obtained.” A more broad definition is supplied by James, [35], as “any technique making use of random numbers to solve a problem.” According to Wikraynaratna (2000), a more encompassing description is due to Niederreiter [60]: Monte Carlo method is “a numerical method based on random sampling.” Emam et al. [20] note: “According to recent survey on the solution methods of nonlinear stochastic dynamics (Schuëller et al., [74]; Computational Stochastic Mechanics, [73]), it has been shown that the available analytical procedures have remarkable restrictions with respect to dimensionality and type of nonlinearity. Most of the analytical methods are confined to International Applied Mechanics, Vol. 39, No. 7, 2003 1063-7095/03/3907-0753$25.00 ©2003 Plenum Publishing Corporation 753 Florida Atlantic University, Boca Raton, USA. Published in Prikladnaya Mekhanika, Vol. 39, No. 7, pp. 3–14, July 2003. Original article submitted April 22, 2003.