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Sawilowsky [56] distinguishes between a simulation, a Monte Carlo method, and a Monte Carlo simulation: a simulation is a fictitious representation of reality, a Monte Carlo method is a technique that can be used to solve a mathematical or statistical problem, and a Monte Carlo simulation uses repeated sampling to obtain the statistical ...
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.
The variance of randomly generated points within a unit square can be reduced through a stratification process. In mathematics, more specifically in the theory of Monte Carlo methods, variance reduction is a procedure used to increase the precision of the estimates obtained for a given simulation or computational effort. [1]
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 .
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.
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 .
As it is commonly the case for Monte-Carlo methods, this algorithm performs particularly well when the dimension is higher than , and one only needs a small set of values. Indeed, the computational cost increases linearly with the dimension, whereas the cost of grid dependant methods increase exponentially with the dimension.
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.