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J. F. Reis et al.
1.5 Monte Carlo Methods
Monte Carlo (MC) methods are a broad class of algorithms that find their application
in a wide variety of engineering fields. For instance, they are exploited to solve
optimisation problems; they can be used to compute complex integrals, or they can
be used to generate draws from a probability distribution.
The MC methods rely on a repeated random sampling to realise a large set of
numerical experiments to obtain statistical information about the stochastic process.
The main idea onto which a MC method is built consists in using randomness to
solve a problem that might be deterministic in principle. In particular, the law of
large numbers allows us to compute a QoI in the statistical sense. For instance, the
expected value of a random variable, like the probability of getting head or tail when
tossing a coin, may be estimated by simply running a large number of independent
experiments (or realisations).
Given their simplicity, MC methods are often used as a brute-force approach to
tackle problems that otherwise would be too difficult, or even impossible, to solve.
Nowadays, there exists a broad family of MC algorithms that are used in science and
engineering; we recall here the importance sampling often employed in statistical
physics, the direct simulation Monte Carlo (DSMC) used in micro-fluidics problems
or the Monte Carlo localisation (MCL) applied in autonomous robotics.
1.5.1 Mean and Variance
The goal of this section is to show how two fundamental QoIs, the mean and
the variance of the model output, can be computed using an MC approach. We
now focus on the heat diffusion equation (Eq. (1.15)) where the uncertainty is
related to the thermal conductivity; k is parametrised using the KL method. Since
a stochastic field is involved, an infinite number of parameters should be used in
this parametrisation. As shown in Sect. 1.4.1, we are sometimes allowed to truncate
the KL series to reduce the dimensions of the stochastic space. In particular, in the
limit of a very large correlation length, we could parametrise the whole field with
respect of a one single random variable, i.e. we could truncate the series at the first
term. Otherwise, we would have faced a slower eigenvalue decay, and, as a result,
we should have included a larger number, though still finite, of terms within the
truncated KL expansion. Nevertheless, the procedure to compute the mean and the
variance would essentially be the same.
The deterministic solution of the heat diffusion equation is found by assigning
specific values of k i at each x i element employed in the Galerkin finite element
approach. As reported in Algorithm 1, the KL expansion can be exploited to
generate a set of N el correlated k i samples, starting from the germ. As the domain
is represented through a finite discretisation, the dimension of the germ is at most
J. F. Reis et al.
1.5 Monte Carlo Methods
Monte Carlo (MC) methods are a broad class of algorithms that find their application
in a wide variety of engineering fields. For instance, they are exploited to solve
optimisation problems; they can be used to compute complex integrals, or they can
be used to generate draws from a probability distribution.
The MC methods rely on a repeated random sampling to realise a large set of
numerical experiments to obtain statistical information about the stochastic process.
The main idea onto which a MC method is built consists in using randomness to
solve a problem that might be deterministic in principle. In particular, the law of
large numbers allows us to compute a QoI in the statistical sense. For instance, the
expected value of a random variable, like the probability of getting head or tail when
tossing a coin, may be estimated by simply running a large number of independent
experiments (or realisations).
Given their simplicity, MC methods are often used as a brute-force approach to
tackle problems that otherwise would be too difficult, or even impossible, to solve.
Nowadays, there exists a broad family of MC algorithms that are used in science and
engineering; we recall here the importance sampling often employed in statistical
physics, the direct simulation Monte Carlo (DSMC) used in micro-fluidics problems
or the Monte Carlo localisation (MCL) applied in autonomous robotics.
1.5.1 Mean and Variance
The goal of this section is to show how two fundamental QoIs, the mean and
the variance of the model output, can be computed using an MC approach. We
now focus on the heat diffusion equation (Eq. (1.15)) where the uncertainty is
related to the thermal conductivity; k is parametrised using the KL method. Since
a stochastic field is involved, an infinite number of parameters should be used in
this parametrisation. As shown in Sect. 1.4.1, we are sometimes allowed to truncate
the KL series to reduce the dimensions of the stochastic space. In particular, in the
limit of a very large correlation length, we could parametrise the whole field with
respect of a one single random variable, i.e. we could truncate the series at the first
term. Otherwise, we would have faced a slower eigenvalue decay, and, as a result,
we should have included a larger number, though still finite, of terms within the
truncated KL expansion. Nevertheless, the procedure to compute the mean and the
variance would essentially be the same.
The deterministic solution of the heat diffusion equation is found by assigning
specific values of k i at each x i element employed in the Galerkin finite element
approach. As reported in Algorithm 1, the KL expansion can be exploited to
generate a set of N el correlated k i samples, starting from the germ. As the domain
is represented through a finite discretisation, the dimension of the germ is at most