Chapter 4
Monte Carlo Computing
We summarize the basic elements of the Monte Carlo method for solving integral
equations. The method extensively employs concepts and notions of probability
theory and statistics. They are marked in italic and defined in the appendix in the
case that the reader wants to refresh his insight on the subject during reading this
section. Where possible, we tried to synchronize the notations used in the probability
theory with these used in the here presented Monte Carlo approaches.
4.1 The Monte Carlo Method for Solving Integrals
The expectation value E{x }= E
x
of a random variable x , which takes values x(Q)
with a probability density p
ψ
(Q), is given by the integral
E
x
=
dQp
x
(Q)x(Q), (4.1)
where Q ∈ R
n
can be a multi-dimensional point or vector. The fundamental Monte
Carlo method for evaluation of E
x
is to carry out N independent experiments,
or applications of the probability density p
x
, which generate N random points
Q
1
,...,Q
N
, which represent an N -dimensional statistical sample of x . The sample
mean η is related to the expectation value E
x
by
E
x
≃ η =
1
N
N
i =1
x(Q
i
), P {|E
x
− η|≤
3σ
x
√
N
}≃ 0.997 , (4.2)
with a precision, which depends on the number of independent applications N and
the standard deviation σ
x
of the random variable. According to the three σ rule,
© Springer Nature Switzerland AG 2021
M. Nedjalkov et al., Stochastic Approaches to Electron Transport
in Micro- and Nanostructures, Modeling and Simulation in Science,
Engineering and Technology, https://doi.org/10.1007/978-3-030-67917-0_4
39