Gradient-enhanced kriging (GEK) is a surrogate modeling technique used in engineering. A surrogate model (alternatively known as a metamodel, response surface or emulator) is a prediction of the output of an expensive computer code. This prediction is based on a small number of evaluations of the expensive computer code.
Introduction
Adjoint solvers are now becoming available in a range of computational fluid dynamics (CFD) solvers, such as Fluent, OpenFOAM, SU2 and US3D. Originally developed for optimization, adjoint solvers are now finding more and more use in uncertainty quantification.
Linear speedup An adjoint solver allows one to compute the gradient of the quantity of interest with respect to all design parameters at the cost of one additional solve. This, potentially, leads to a linear speedup: the computational cost of constructing an accurate surrogate decrease, and the resulting computational speedup s {\displaystyle s} scales linearly with the number d {\displaystyle d} of design parameters. The reasoning behind this linear speedup is straightforward. Assume we run N {\displaystyle N} primal solves and N {\displaystyle N} adjoint solves, at a total cost of 2 N {\displaystyle 2N} . This results in N + d N {\displaystyle N+dN} data; N {\displaystyle N} values for the quantity of interest and d {\displaystyle d} partial derivatives in each of the N {\displaystyle N} gradients. Now assume that each partial derivative provides as much information for our surrogate as a single primal solve. Then, the total cost of getting the same amount of information from primal solves only is N + d N {\displaystyle N+dN} . The speedup is the ratio of these costs:
s = N + d N 2 N = 1 2 + 1 2 d . {\displaystyle s={\frac {N+dN}{2N}}={\frac {1}{2}}+{\frac {1}{2}}d.}
A linear speedup has been demonstrated for a fluid-structure interaction problem and for a transonic airfoil.
Noise One issue with adjoint-based gradients in CFD is that they can be particularly noisy. When derived in a Bayesian framework, GEK allows one to incorporate not only the gradient information, but also the uncertainty in that gradient information.
Approach When using GEK one takes the following steps:
Create a design of experiment (DoE): The DoE or 'sampling plan' is a list of different locations in the design space. The DoE indicates which combinations of parameters one will use to sample the computer simulation. With Kriging and GEK, a common choice is to use a Latin Hypercube Design (LHS) design with a 'maximin' criterion. The LHS-design is available in scripting codes like MATLAB or Python. Make observations: For each sample in our DoE one runs the computer simulation to obtain the Quantity of Interest (QoI). Construct the surrogate: One uses the GEK predictor equations to construct the surrogate conditional on the obtained observations. Once the surrogate has been constructed it can be used in different ways, for example for surrogate-based uncertainty quantification (UQ) or optimization.
Predictor equations In a Bayesian framework, we use Bayes' theorem to predict the Kriging mean and covariance conditional on the observations. When using GEK, the observations are usually the results of a number of computer simulations. GEK can be interpreted as a form of Gaussian process regression.
Kriging Along the lines of, we are interested in the output X {\displaystyle X} of our computer simulation, for which we assume the normal prior probability distribution:
X ∼ N ( μ , P ) , {\displaystyle X\sim {\mathcal {N}}(\mu ,P),}
with prior mean μ {\displaystyle \mu } and prior covariance matrix P {\displaystyle P} . The observations y {\displaystyle y} have the normal likelihood:
Y ∣ x ∼ N ( H x , R ) , {\displaystyle Y\mid x\sim {\mathcal {N}}(Hx,R),}
with H {\displaystyle H} the observation matrix and R {\displaystyle R} the observation error covariance matrix, which contains the observation uncertainties. After applying Bayes' theorem we obtain a normally distributed posterior probability distribution, with Kriging mean:
E ( X ∣ y ) = μ + K ( y − H μ ) , {\displaystyle \operatorname {E} (X\mid y)=\mu +K(y-H\mu ),}
and Kriging covariance:
cov ( X ∣ y ) = ( I − K H ) P , {\displaystyle \operatorname {cov} (X\mid y)=(I-KH)P,}
where we have the gain matrix:
K = P H T ( R + H P H T ) − 1 . {\displaystyle K=PH^{T}(R+HPH^{T})^{-1}.}
… excerpt ends here. Continue reading the full article.
