The Monte Carlo Method

The Monte Carlo method is a numerical method of solving mathematical problems by the simulation of random variables. It performs simulation of any process whose development is influenced by random factors, but also if the given problem involves no chance, the method enables artificial construction of a probabilistic model.

The generally accepted birth date of this method is 1949 when an article entitled "The Monte Carlo method" by Metropolis and Ulam appeared. Curiously enough, the theoretical foundation of the method had been known long before first articles were published. Well before 1949 certain problems in statistics were sometimes solved by means of random sampling - that is, in fact, by the Monte Carlo method. However, because simulation of random variables by hand is a laborious process, the use of the Monte Carlo method as a universal numerical technique became practical only with the advent of computers and high-quality pseudorandom number generators.

Monte Carlo is one of the most versatile and widly used numerical methods, unbeatable for solving multidimensional problems in composite domains. However, the Monte Carlo method can be very slow. That is why much of the effort in development of Monte Carlo has been in construction of variance reduction methods which speed up the computation.

While numerical integration remains a major Monte Carlo application, the MCM is used for a staggering number of applications. A large class of other MCM applications can be described as MCMs based on the construction of Markov chains as the probability distribution used for calculating expected values. These include methods based on both discrete and continuous Markov chains. These MCMs are used to solve problems in transport theory, nuclear medicine, computer graphics, finance, biophysics, porous media, computational chemistry, materials science and nuclear weapons design.