Monte Carlo Analysis
Introduction
The Monte Carlo method[1] is a non-deterministic or numerical statistical method, used to approximate complex mathematical expressions that are expensive to evaluate accurately. The method was named in reference to the Monte Carlo Casino (Monaco) for being “the capital of the game of chance”, as roulette is a simple generator of random numbers. The name and systematic development of Monte Carlo methods dates back to approximately 1944 and was greatly improved with the development of the computer.
The use of Monte Carlo methods in scientific research originated during the development of the atomic bomb in World War II, within the Los Alamos National Laboratory in the United States. In this context, the researchers used probabilistic simulations to study hydrodynamic phenomena related to the diffusion of neutrons in fission materials, an inherently random behavioral process.
Currently, Monte Carlo methods are applied in a wide variety of fields, from nuclear physics and computational statistics to computer graphics, where they form the basis of numerous ray tracing algorithms used for the generation of realistic three-dimensional images.
In the first stage of these investigations, John von Neumann and Stanislaw Ulam refined this roulette and the methods of division of tasks. However, the systematic development of these ideas had to wait for the work of Harris and Herman Kahn in 1948. In approximately the same year, Enrico Fermi, Nicholas Metropolis and Ulam obtained estimators for the characteristic values of the Schrödinger equation for neutron capture at the nuclear level using this method.
The Monte Carlo method provides approximate solutions to a wide variety of mathematical problems, making it possible to carry out experiments with sampling of pseudorandom numbers on a computer. The method is applicable to any type of problem, whether stochastic or deterministic. Unlike numerical methods that rely on evaluations at N points in an M-dimensional space to produce an approximate solution, the Monte Carlo method has an absolute error of estimation that decreases as under the central limit theorem.
Origins of the method
The invention of the Monte Carlo method is attributed to Stanislaw Ulam and John von Neumann. Ulam has explained how he came up with the idea while playing solitaire during an illness in 1946. He noted that it is much simpler to get an idea of the overall result of solitaire by trying multiple cards and counting the proportions of the results than it is to compute all combination possibilities formally. It occurred to him that this same observation should apply to his Los Alamos work on neutron diffusion, for which it is virtually impossible to solve the integral-differential equations governing scattering, absorption, and fission. In his words, "the idea was to test the thousands of possibilities with mental experiments, and at each stage, determine by chance, by a random number distributed according to probabilities, what would happen and total all the possibilities and have an idea of the behavior of the physical process."