14 Best Practices for Surrogate Based Uncertainty Quantification in. . .
437
Table 14.1 Suitability of surrogate models for Uncertainty Quantification
Surrogate model
Scattered data Error available Use of gradient
Convenient for high
dimensions
Kriging
Y
Y
Y
Y
Radial basis functions Y
N
N
Y
Spline
Y
N
Y
N
Polynomial regression Y
N
Y
N
Sparse grid
N
N
N
N
input uncertainties [21]. This approach focuses on regions that are often sampled
through Monte Carlo (addition of input PDFs) and have a high surrogate error.
This refinement strategy is in line with the DoE sampling strategy of adding more
points in regions that are sampled more often, as eventually the surrogate error is
proportional to the distance among the points.
14.3.2 Advantages of Surrogate Modelling for Uncertainty
Quantification
Following [10], the engineering requirements for an effective uncertainty quantification technique are:
• Require no more than 100 evaluations of the full order model,
• Scattered evaluation positions can be obtained,
• Possibility to incorporate pre-existing evaluations of the quantity of interest in
the uncertainty quantification,
• Robustness towards failure in the evaluation of the objective function (the sample
points should not be fixed). This is cannot be possible with fixed integration rules
where the sampling distribution is fixed beforehand,
• Availability of the exploitation of error estimation and error reduction techniques,
• Possibility to extend to high dimensional problems.
Based on these points, it is possible to construct Table 14.1 with the suitability
of the different surrogate models for Uncertainty Quantification with a special focus
on aerodynamic shape optimization problems:
According to these requirements, surrogate methods are an attractive approach
for the efficient quantification using CFD. Among the different surrogates, Kriging
has become one of the most used ones in engineering [19]. It follows a probabilistic
framework according to Bayesian statistics, featuring a built-in measure of the
prediction error [19, 22]. In contrast with Polynomial Chaos Expansion, Kriging
interpolation works well with nonlinear functions such as the drag coefficient.
Finally, by using surrogate models, arbitrary locations of the samples can be chosen
in contrast of sparse grid techniques.
437
Table 14.1 Suitability of surrogate models for Uncertainty Quantification
Surrogate model
Scattered data Error available Use of gradient
Convenient for high
dimensions
Kriging
Y
Y
Y
Y
Radial basis functions Y
N
N
Y
Spline
Y
N
Y
N
Polynomial regression Y
N
Y
N
Sparse grid
N
N
N
N
input uncertainties [21]. This approach focuses on regions that are often sampled
through Monte Carlo (addition of input PDFs) and have a high surrogate error.
This refinement strategy is in line with the DoE sampling strategy of adding more
points in regions that are sampled more often, as eventually the surrogate error is
proportional to the distance among the points.
14.3.2 Advantages of Surrogate Modelling for Uncertainty
Quantification
Following [10], the engineering requirements for an effective uncertainty quantification technique are:
• Require no more than 100 evaluations of the full order model,
• Scattered evaluation positions can be obtained,
• Possibility to incorporate pre-existing evaluations of the quantity of interest in
the uncertainty quantification,
• Robustness towards failure in the evaluation of the objective function (the sample
points should not be fixed). This is cannot be possible with fixed integration rules
where the sampling distribution is fixed beforehand,
• Availability of the exploitation of error estimation and error reduction techniques,
• Possibility to extend to high dimensional problems.
Based on these points, it is possible to construct Table 14.1 with the suitability
of the different surrogate models for Uncertainty Quantification with a special focus
on aerodynamic shape optimization problems:
According to these requirements, surrogate methods are an attractive approach
for the efficient quantification using CFD. Among the different surrogates, Kriging
has become one of the most used ones in engineering [19]. It follows a probabilistic
framework according to Bayesian statistics, featuring a built-in measure of the
prediction error [19, 22]. In contrast with Polynomial Chaos Expansion, Kriging
interpolation works well with nonlinear functions such as the drag coefficient.
Finally, by using surrogate models, arbitrary locations of the samples can be chosen
in contrast of sparse grid techniques.