438
C. Sabater
14.3.3 Gradient-Enhanced Surrogates for Efficient UQ
When dealing with optimization problems, one of the benefits of surrogate modelling is that the gradients of the objective function are not required. Having
available the gradients, it is possible to directly choose a gradient descent approach
[23]. However, if a global search is desired, the addition of the gradients can enhance
the accuracy of the surrogate model. In the case of surrogate based uncertainty
quantification this is crucial, as the addition of gradients can improve the global
representation of the design or uncertain landscape for a more accurate integration
through classical Monte Carlo Methods.
The main drawback of gradient-enhanced surrogate models is the increase in
complexity of the model and the increase in size of the correlation matrix, leading
to lengthier surrogate parameter estimation. This is the cost to pay in order to have
more accurate predictions.
14.3.3.1 Gradient-Enhanced Kriging
The use of Gradient-Enhanced Kriging (GEK) [19] is especially useful when a
large number of uncertain variables is present. Following [10, 20], it is shown that
employing GEK using gradient information that is cheaply evaluated, is as effective
as common integration methods such as probabilistic collocation and sparse grid
techniques. With the increase in dimensionality, it is expected that GEK will be
more efficient, as the cost of the gradients is kept constant. The increase in accuracy
obtained with GEK w.r.t. Kriging is shown in Fig. 14.1 for an analytical function
with 15 samples following Sobol sequences. By obtaining the gradients in addition
to the function value, the representation of the landscape is more accurate.
The gradients must only be added to the surrogate if these can be cheaply
evaluated. Otherwise, it is more useful to enhance the meta-model by simply
evaluating more points of the full order model. The use of traditional methods to
obtain the gradients (finite differences or complex step differentiation) should be
then substituted by others more advanced such as the adjoint method or algorithmic
differentiation.
14.3.3.2 The Adjoint Method: Breaking the Curse of Dimensionality
The adjoint method computes the partial gradients with respect to all design
parameters or uncertainties at the cost of about one flow solution; in other words, the
cost of the adjoint method is independent of the number of parameters. It provides
an effective method to build the Gradient-Enhanced Kriging model of the response
surface.
C. Sabater
14.3.3 Gradient-Enhanced Surrogates for Efficient UQ
When dealing with optimization problems, one of the benefits of surrogate modelling is that the gradients of the objective function are not required. Having
available the gradients, it is possible to directly choose a gradient descent approach
[23]. However, if a global search is desired, the addition of the gradients can enhance
the accuracy of the surrogate model. In the case of surrogate based uncertainty
quantification this is crucial, as the addition of gradients can improve the global
representation of the design or uncertain landscape for a more accurate integration
through classical Monte Carlo Methods.
The main drawback of gradient-enhanced surrogate models is the increase in
complexity of the model and the increase in size of the correlation matrix, leading
to lengthier surrogate parameter estimation. This is the cost to pay in order to have
more accurate predictions.
14.3.3.1 Gradient-Enhanced Kriging
The use of Gradient-Enhanced Kriging (GEK) [19] is especially useful when a
large number of uncertain variables is present. Following [10, 20], it is shown that
employing GEK using gradient information that is cheaply evaluated, is as effective
as common integration methods such as probabilistic collocation and sparse grid
techniques. With the increase in dimensionality, it is expected that GEK will be
more efficient, as the cost of the gradients is kept constant. The increase in accuracy
obtained with GEK w.r.t. Kriging is shown in Fig. 14.1 for an analytical function
with 15 samples following Sobol sequences. By obtaining the gradients in addition
to the function value, the representation of the landscape is more accurate.
The gradients must only be added to the surrogate if these can be cheaply
evaluated. Otherwise, it is more useful to enhance the meta-model by simply
evaluating more points of the full order model. The use of traditional methods to
obtain the gradients (finite differences or complex step differentiation) should be
then substituted by others more advanced such as the adjoint method or algorithmic
differentiation.
14.3.3.2 The Adjoint Method: Breaking the Curse of Dimensionality
The adjoint method computes the partial gradients with respect to all design
parameters or uncertainties at the cost of about one flow solution; in other words, the
cost of the adjoint method is independent of the number of parameters. It provides
an effective method to build the Gradient-Enhanced Kriging model of the response
surface.