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Gradient-enhanced kriging
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Gradient-enhanced kriging
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.
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.
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 scales linearly with the number of design parameters.
The reasoning behind this linear speedup is straightforward. Assume we run primal solves and adjoint solves, at a total cost of . This results in data; values for the quantity of interest and partial derivatives in each of the 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 . The speedup is the ratio of these costs:
A linear speedup has been demonstrated for a fluid-structure interaction problem and for a transonic airfoil.
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.
When using GEK one takes the following steps:
Once the surrogate has been constructed it can be used in different ways, for example for surrogate-based uncertainty quantification (UQ) or optimization.
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Gradient-enhanced kriging
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.
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.
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 scales linearly with the number of design parameters.
The reasoning behind this linear speedup is straightforward. Assume we run primal solves and adjoint solves, at a total cost of . This results in data; values for the quantity of interest and partial derivatives in each of the 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 . The speedup is the ratio of these costs:
A linear speedup has been demonstrated for a fluid-structure interaction problem and for a transonic airfoil.
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.
When using GEK one takes the following steps:
Once the surrogate has been constructed it can be used in different ways, for example for surrogate-based uncertainty quantification (UQ) or optimization.