Monte Carlo Methods
The written history of the Monte Carlo methods began in 1949 after the paper of Metropolis and Ulm , but at that time the method had already been used for several years in secret defense projects of the United States of America for simulating the behaviour of nuclear reactors. It was definitely known to J. Von Neumann, N. Metropolis, S.M. Ulm, H. Kahn and their coworkers at the Los Alamos Scientific Laboratory much before its publication. The name “Monte Carlo” comes from the random or chance character of the method and the famous casino in Monaco (South of France). It may be observed that finding random numbers is a crucial step of the Monte Carlo methods. The numbers which are generated at the roulette table in Monte Carlo are truly random, but the numbers which are actually utilized in Monte Carlo methods are generated on the computer using deterministic formulae which are called pseudo-random numbers or quasi-random numbers. A fundamental difficulty of the Monte Carlo method stems from the requirement that the nodes be independent random samples. The problem arises for generating independent random samples concretely. The users of Monte Carlo methods avoid this problem by using pseudo-random numbers in place of truly random samples. The first paper with the term quasi-Monte Carlo came to light by the Tech-report in 1951 by Richtmyer. This method can be described in a simple way as the deterministic version of a Monte Carlo method in the sense that the random samples in the Monte Carlo method are replaced by well-selected deterministic points.
KeywordsMonte Carlo Method Quadrature Rule Maxwellian Distribution Particle Method Star Discrepancy
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