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C. Sabater
14.3.1.1 Design of Experiments
One of the key elements for the construction of surrogate models is the initial
sampling strategy, the Design of Experiments (DoE). Both for optimization and
uncertainty quantification, the choice of sampling strategy must follow from:
• The design landscape is fully explored.
• The sampling must follow minimum discrepancy, i.e. the sampling plan must be
based on uniform partitions of the unit interval but not regular.
• The projections of the sampling points to the axis of each variable are uniform.
• The number of expensive simulations/full order model evaluations to achieve this
complete exploration should be minimum.
• The number of samples should not be fixed. The addition of more points can be
used to further refine the surrogate without the need of recomputing the previous
ones.
One of the most common sampling approaches is the use of Latin Hypercube
Sampling developed by Mckay [12]. However, Quasi Monte Carlo techniques that
employ deterministic low discrepancy sequences [13] can prove to be more effective
to obtain an even sampling distribution [14].
A Sobol Sequence is a low discrepancy, quasi-random sequence that uses a base
of two to successively create fined uniform partitions [15]. As a result, it is possible
to improve its accuracy as more samples are added, and it is possible to reuse the
existing points.
The surrogate accuracy can be improved for uncertainty quantification if the DoE
follows the distribution of the input uncertainties [7, 16, 17]. Sampling points should
then be added in areas that are prone to be sampled along the mean, rather than
in the tails of the input uncertainties. When direct integration is performed on the
surrogate, the accuracy will increase in regions that were sampled more often.
14.3.1.2 Refinement Strategy for Uncertainty Quantification
Commonly, in surrogate based optimization, the refinement strategy follows an
equilibrium between exploration and exploitation. In the first one, the location of
the maximum expected improvement [18] or maximum surrogate error is sampled.
In the latter stage, regions around the current optimum are investigated by validating
the surrogate optimum in the full order model or through trust-region methods. For
a more details on a refinement strategy for optimization, refer to [19].
In a similar manner, it is possible to refine surrogate models to be applied in
uncertainty quantification. In case the objective function follows Eq. (14.8), the
reliability-based approach, refinement along the 95% quantile can be achieved as
in standard optimization, by means of expected improvement [2, 20].
Otherwise, the refinement strategy follows from a global exploration of the
uncertain landscape. The most common criteria is sampling the location with the
maximum product of the surrogate error with the probability density function of the
C. Sabater
14.3.1.1 Design of Experiments
One of the key elements for the construction of surrogate models is the initial
sampling strategy, the Design of Experiments (DoE). Both for optimization and
uncertainty quantification, the choice of sampling strategy must follow from:
• The design landscape is fully explored.
• The sampling must follow minimum discrepancy, i.e. the sampling plan must be
based on uniform partitions of the unit interval but not regular.
• The projections of the sampling points to the axis of each variable are uniform.
• The number of expensive simulations/full order model evaluations to achieve this
complete exploration should be minimum.
• The number of samples should not be fixed. The addition of more points can be
used to further refine the surrogate without the need of recomputing the previous
ones.
One of the most common sampling approaches is the use of Latin Hypercube
Sampling developed by Mckay [12]. However, Quasi Monte Carlo techniques that
employ deterministic low discrepancy sequences [13] can prove to be more effective
to obtain an even sampling distribution [14].
A Sobol Sequence is a low discrepancy, quasi-random sequence that uses a base
of two to successively create fined uniform partitions [15]. As a result, it is possible
to improve its accuracy as more samples are added, and it is possible to reuse the
existing points.
The surrogate accuracy can be improved for uncertainty quantification if the DoE
follows the distribution of the input uncertainties [7, 16, 17]. Sampling points should
then be added in areas that are prone to be sampled along the mean, rather than
in the tails of the input uncertainties. When direct integration is performed on the
surrogate, the accuracy will increase in regions that were sampled more often.
14.3.1.2 Refinement Strategy for Uncertainty Quantification
Commonly, in surrogate based optimization, the refinement strategy follows an
equilibrium between exploration and exploitation. In the first one, the location of
the maximum expected improvement [18] or maximum surrogate error is sampled.
In the latter stage, regions around the current optimum are investigated by validating
the surrogate optimum in the full order model or through trust-region methods. For
a more details on a refinement strategy for optimization, refer to [19].
In a similar manner, it is possible to refine surrogate models to be applied in
uncertainty quantification. In case the objective function follows Eq. (14.8), the
reliability-based approach, refinement along the 95% quantile can be achieved as
in standard optimization, by means of expected improvement [2, 20].
Otherwise, the refinement strategy follows from a global exploration of the
uncertain landscape. The most common criteria is sampling the location with the
maximum product of the surrogate error with the probability density function of the