Polynomial chaos expansions and stochastic finite-element methods - Risk and Reliability in Geotechnical Engineering - page 263

Step B Step A Step C

Quantification of Model(s) of the Uncertainty propagation

sources of uncertainty assessment criteria

Computational model

Moments

Random variables

Probability of failure

Response PDF

Step C´

Sensitivity analysis

Figure 6.1 Uncertainty quantification framework.

whose properties are implicitly defined by the propagation of the uncertainties described
by the joint distribution f X through the computational model (Step C). This step consists of
characterizing the probabilistic content of Y , that is, its statistical moments, quantiles, or
full distribution, to derive confidence intervals (CIs) around the mean QoI for robust predic-
tions, or to carry out reliability assessment.
When the spatial variability of soil properties is to be modeled, random fields have to
be used. The mathematical description of random fields and their discretization is beyond
the scope of this chapter. For an overview, the interested reader is referred to Vanmarcke
(1983); Sudret and Der Kiureghian (2000); and Sudret and Berveiller (2007). In any case,
and whatever the random field discretization technique (e.g., Karhunen-Love [KL] expan-
sion, expansion optimal linear estimation [EOLE], etc.), the problem eventually reduces to
an input random vector (which is usually Gaussian) appearing in the discretization. Then
the uncertainty propagation issue identically suits the framework described above. For the
sake of illustration, an application example involving spatial variability will be addressed
in Section 6.6.3.

6.2.2 Monte Carlo simulation
The Monte Carlo simulation (MCS) is a well-known technique for estimating statisti-
cal properties of the random response Y = M( X ): realizations of the input vector X are
sampled according to the input distribution f X , then the computational model M is run for
each sample, and the resulting set of QoI is post-processed (Rubinstein and Kroese, 2008).
Although rather universal, MCS suffers from low efficiency. Typically, 10 3−4 samples are
required to reach an acceptable accuracy. The cost even blows up when probabilities of
failure are to be computed for the sake of reliability assessment, since 10 k +2 samples are
required when estimating a probability of 10 − k . Thus, alternative methods have to be
devised for addressing uncertainty quantification problems that involve computationally
demanding models such as finite-element models. In the past decade, PCEs have become a
popular approach in this respect.