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Monte Carlo methods vary, but tend to follow a particular pattern: Define a domain of possible inputs; Generate inputs randomly from a probability distribution over the domain; Perform a deterministic computation of the outputs; Aggregate the results; Monte Carlo method applied to approximating the value of π
An illustration of Monte Carlo integration. In this example, the domain D is the inner circle and the domain E is the square. Because the square's area (4) can be easily calculated, the area of the circle (π*1.0 2) can be estimated by the ratio (0.8) of the points inside the circle (40) to the total number of points (50), yielding an approximation for the circle's area of 4*0.8 = 3.2 ≈ π.
This Monte Carlo method is independent of any relation to circles, and is a consequence of the central limit theorem, discussed below. These Monte Carlo methods for approximating π are very slow compared to other methods, and do not provide any information on the exact number of digits that are obtained.
A Monte Carlo simulation shows a large number and variety of possible outcomes, including the least likely as well … Continue reading → The post Understanding How the Monte Carlo Method Works ...
The general motivation to use the Monte Carlo method in statistical physics is to evaluate a multivariable integral. The typical problem begins with a system for which the Hamiltonian is known, it is at a given temperature and it follows the Boltzmann statistics .
This can be used to design a Monte Carlo method for approximating the number π, although that was not the original motivation for de Buffon's question. [3] The seemingly unusual appearance of π in this expression occurs because the underlying probability distribution function for the needle orientation is rotationally symmetric.
Like in any other Monte Carlo method, there are correlations of the samples being drawn from (). A typical measurement of the correlation is the tunneling time . The tunneling time is defined by the number of Markov steps (of the Markov chain) the simulation needs to perform a round-trip between the minimum and maximum of the spectrum of F .
The antithetic variates technique consists, for every sample path obtained, in taking its antithetic path — that is given a path {, …,} to also take {, …,}.The advantage of this technique is twofold: it reduces the number of normal samples to be taken to generate N paths, and it reduces the variance of the sample paths, improving the precision.