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Adaptive Monte Carlo integration package Vegas Importance sampling Monte Carlo phase space integration Monte Carlo phase space integration Contents


Adaptive Monte Carlo integration package Vegas

This section contains a short description of the adaptive Monte Carlo program VEGAS. See for details [Lepage-1978,Press-1992].

The Monte Carlo method reduces a task of integral evaluation to the task of mean value calculation. Let $g(x)$ is a density function satisfying


\begin{displaymath}\int \! g(x) \,dx = 1,\end{displaymath}

then

\begin{displaymath}\int \! f(x) \, dx = \int \! f(x)/g(x) \, \,g(x) \, dx = \; <\!\!f/g\!\!> \; = \lim_{ N \to \infty} \sum (f(x_i)/g(x_i)) / N, \end{displaymath}

where points $x_i$ are sampled with the probability density $g(x) \, dx.$

The uncertainty $\sigma_N$ of $<\!f/g\!>$ estimation by $N$ sample points is proportional to square root of function's variance divided over $N$:

\begin{displaymath}\sigma_N = \sqrt{ (<\!\!(f/g)^2\!\!> - <\!\!f/g\!\!>^2) / N }\;. \end{displaymath}

VEGAS uses two techniques which allow to decrease the uncertainty of Monte Carlo calculation, namely the importance sampling and the stratified sampling.


Subsections