Environmental Engineering Reference
In-Depth Information
2.3 The Monte Carlo method
2.3.1 The general characteristics of the Monte Carlo
method
The Monte-Carlo metods is the common name of the group of numerical
methods based on obtaining a large number of realisations of the stochastic
(random) process, which is formed so that its probability characteristics
coincide with similar values of the given task. The methods are used to solve
problems in physics, mathematics, economics, optimisation, control theory,
etc.
27,etc
In recent years the methods have been applied in several studies
to ensure the reliability of NPP equipment and piping and maintenance
optimisation
29, etc
.
The principle of the Monte Carlo method is as follows: find the value
a
of some quantity to be investigated. To do this, choose a random variable
X
whose mathematical expectation is equal to
a
:
M
(
X
) =
a
.
In practice, the following procedure is used: carry out
n
tests which results
in
n
possible values of
X
, compute their arithmetic mean and take
x xn
= S
(
)/
i
as an estimate (approximate value)
a
* of the unknown number
a
:
*
≈=
Since the Monte Carlo method requires a large number of tests, it is often
called the method of statistical tests. The theory of this method indicates how
to choose random variable
X
, how to find its possible values. In particular,
procedures for reducing the variance of random variables are developed,
resulting in reduced errors in replacing the unknown mathematical expectation
a
by its estimate
a
*.
aa x
.
2.3.2
The procedure for estimating the inaccuracy of the
Monte Carlo method
Suppose that
n
independent trials have been performed to estimating
a
*
expectation as a random variable
X
has been made (
n
possible values of
X
were
used) and as a results the sample mean
x
was found which is accepted
as the desired estimate
a
*=
x
. It is clear that if the experiments are repeated,
other possible values of
X
are obtained and, therefore, another mean and hence
a different estimate of
a
*. This already implies that an exact estimate of the
mathematical expectation is not impossible. The question naturally arises about
the value of the permissible error. We restrict ourselves to finding only the
upper limit δ of the permissible error with a given probability (reliability) γ:
(
PX
− ≤δ =g
.
The upper limit of error δ in which we are interested is in fact the
accuracy estimate of the mathematical expectation of an average sample using
confidence intervals. The following three cases will be considered.
)
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