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5.1 Monte Carlo integration
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The most common use of Monte Carlo methods is the evaluation of multidimensional integrals [24]. Consider first the approximation of an
integral obtained from the trapezoidal rule (3.3#eq.1)
Instead of a uniform sampling, imagine an evaluation where the positions
are random numbers uniformly distributed in the
interval
; this yields a Monte Carlo integration

(2) 
The convergence rate is of course lower than from a uniform sampling
(5.1#eq.1); the strength however appears for
integrals in higher dimensions
, where the Monte Carlo error scales
as
irrespective of the number of dimensions  instead of the
scaling in
that is achieved when using a uniform mesh.
