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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
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A universal augmented framework is proposed in to append derivatives of any order to the observations. This method can be viewed as a generalization of Direct GEK that takes into account higher-order derivatives. Also, the observations and derivatives are not required to be measured at the same
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Current gradient-enhanced kriging methods do not scale well with the number of sampling points due to the rapid growth in the size of the correlation matrix, where new information is added for each sampling point in each direction of the design space. Furthermore, they do not scale well with the
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Example of one-dimensional data interpolated by
Kriging and GEK. The black line indicates the test-function, while the gray circles indicate 'observations', 'samples' or 'evaluations' of the test-function. The blue line is the Kriging mean, the shaded blue area illustrates the Kriging standard
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of 1.25 degrees. We assume that the shape of the airfoil is uncertain; the top and the bottom of the airfoil might have shifted up or down due to manufacturing tolerances. In other words, the shape of the airfoil that we are using might be slightly different from the airfoil that we designed.
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method that maintains accuracy is developed. In addition, this method is able to control the size of the correlation matrix by adding only relevant points defined through the information provided by the partial-least squares method. For more details, see. This approach is implemented into the
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number of independent variables due to the increase in the number of hyperparameters that needs to be estimated. To address this issue, a new gradient-enhanced surrogate model approach that drastically reduced the number of hyperparameters through the use of the
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After designing a sampling plan (indicated by the gray dots) and running the CFD solver at those sample locations, we obtain the
Kriging surrogate model. The Kriging surrogate is close to the reference, but perhaps not as close as we would desire.
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such that it also contains the derivatives (and second derivatives) of the covariance function, see for example . The main advantages of direct GEK over indirect GEK are: 1) we do not have to choose a step-size, 2) we can include
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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
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of the airfoil, based on a large number of CFD simulations. Note that the lowest drag, which corresponds to 'optimal' performance, is close to the undeformed 'baseline' design of the airfoil at (0,0).
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Reference results for the drag coefficient of a transonic airfoil, based on a large number of CFD simulations. The horizontal and vertical axis show the deformation of the shape of the airfoil.
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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
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GEK surrogate model of the drag coefficient of a transonic airfoil. The gray dots indicate the configurations for which the CFD solver was run, the arrows indicate the gradients.
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In the last figure, we have improved the accuracy of this surrogate model by including the adjoint-based gradient information, indicated by the arrows, and applying GEK.
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Han, Z.-H.; Gortz, S.; Zimmermann, R. (2013). "Improving variable-fidelity surrogate modeling via gradient-enhanced kriging and a generalized hybrid bridge function".
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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.
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2014: Uncertainty quantification for the RANS simulation of an airfoil, with the model parameters of the k-epsilon turbulence model as uncertain inputs.
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Chung, H.-S.; Alonso, J.J. (2002). "Using
Gradients to Construct Cokriging Approximation Models for High-Dimensional Design Optimization Problems".
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2015: Uncertainty quantification for the Euler simulation of a transonic airfoil with uncertain shape parameters. Demonstration of a linear speedup.
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Kriging surrogate model of the drag coefficient of a transonic airfoil. The gray dots indicate the configurations for which the CFD solver was run.
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Direct GEK is a form of co-Kriging, where we add the gradient information as co-variables. This can be done by modifying the prior covariance
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of the quantity of interest with respect to all design parameters at the cost of one additional solve. This, potentially, leads to a
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Laurenceau, J.; Sagaut, P. (2008). "Building efficient response surfaces of aerodynamic functions with
Kriging and coKriging".
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deviation. With GEK we can add the gradient information, illustrated in red, which increases the accuracy of the prediction.
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speedup: the computational cost of constructing an accurate surrogate decrease, and the resulting computational speedup
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de Baar, J.H.S.; Scholcz, T.P.; Dwight, R.P. (2015). "Exploiting
Adjoint Derivatives in High-Dimensional Metamodels".
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Zhang, Sheng; Yang, Xiu; Tindel, Samy; Lin, Guang (2021). "Augmented
Gaussian random field: Theory and computation".
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Dwight, R.; Brezillon, J. (2006). "Effect of
Approximations of the Discrete Adjoint on Gradient-Based Optimization".
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2017: Large review of gradient-enhanced surrogate models including many details concerning gradient-enhanced kriging.
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1294:{\displaystyle P_{ij}=\sigma ^{2}\exp \left(-\sum _{k}{\frac {|\xi _{jk}-\xi _{ik}|^{2}}{2\theta _{k}^{2}}}\right),}
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Another way of arriving at the same direct GEK predictor is to append the partial derivatives to the observations
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There are several ways of implementing GEK. The first method, indirect GEK, defines a small but finite stepsize
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is generated from a covariance function. One example of a covariance function is the
Gaussian covariance:
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Once the surrogate has been constructed it can be used in different ways, for example for surrogate-based
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One uses the GEK predictor equations to construct the surrogate conditional on the obtained observations.
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2013: Uncertainty quantification for a transonic airfoil with uncertain angle of attack and Mach number.
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For each sample in our DoE one runs the computer simulation to obtain the
Quantity of Interest (QoI).
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Bouhlel, M.A.; Martins, J.R.R.A. (2018). "Gradient-enhanced kriging for high-dimensional problems".
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de Baar, J.H.S.; Scholcz, T.P.; Verhoosel, C.V.; Dwight, R.P.; van Zuijlen, A.H.; Bijl, H. (2012).
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2329:"Bayesian Design and Analysis of Computer Experiments: Use of Derivatives in Surface Prediction"
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1985:
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1686:), and it runs on Linux, macOS, and Windows. SMT is distributed under the New BSD license.
1865:"Efficient uncertainty quantification with gradient-enhanced Kriging: Applications in FSI"
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2009: Uncertainty quantification for a transonic airfoil with uncertain shape parameters.
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2008: Uncertainty quantification for a transonic airfoil with uncertain shape parameters.
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framework, GEK allows one to incorporate not only the gradient information, but also the
2537:"Restricted-Variance Molecular Geometry Optimization Based on Gradient-Enhanced Kriging"
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Wikle, C.K.; Berliner, L.M. (2007). "A Bayesian tutorial for data assimilation".
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It may require cleanup to comply with Knowledge's content policies, particularly
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1459:, see for example. Indirect Kriging is sensitive to the choice of the step-size
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One issue with adjoint-based gradients in CFD is that they can be particularly
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Raggi, G.; Fdez. Galván, I.; Ritterhoff, C. L.; Vacher, M.; Lindh, R. (2020).
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Ulaganathan, S.; Couckuyt, I.; Dhaene, T.; Degroote, J.; Laermans, E. (2016).
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2030:"Improvements to gradient-enhanced Kriging using a Bayesian interpretation"
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The reasoning behind this linear speedup is straightforward. Assume we run
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technique used in engineering. A surrogate model (alternatively known as a
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Gradient-enhanced kriging for high-dimensional problems (Indirect method)
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2265:"Stochastic Surrogates for Measurements and Computer Models of Fluids"
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2012: Surrogate model construction for a panel divergence problem, a
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Efficient uncertainty quantification using gradient-enhanced Kriging
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1838:"Bayesian design and analysis of computer experiments: two examples"
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516:{\displaystyle s={\frac {N+dN}{2N}}={\frac {1}{2}}+{\frac {1}{2}}d.}
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110:. Statements consisting only of original research should be removed.
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1997:
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and include partial derivative operators in the observation matrix
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2485:"Gradient-Enhanced Universal Kriging for Uncertainty Propagation"
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622:
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2443:"An overview of gradient-enhanced metamodels with applications"
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Laurent, L.; Le Riche, R.; Soulier, B.; Boucard, P.-A. (2017).
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967:{\displaystyle \operatorname {E} (X\mid y)=\mu +K(y-H\mu ),}
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the observation error covariance matrix, which contains the
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2017: Uncertainty propagation for a nuclear energy system.
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Adjoint solvers are now becoming available in a range of
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1993: Design problem for a borehole model test-function.
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194:, adjoint solvers are now finding more and more use in
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2002: Aerodynamic design of a supersonic business jet.
1038:{\displaystyle \operatorname {cov} (X\mid y)=(I-KH)P,}
33:
A major contributor to this article appears to have a
1974:"Algorithm Developments for Discrete Adjoint Methods"
1971:
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Along the lines of, we are interested in the output
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Morris, M.D.; Mitchell, T.J.; Ylvisaker, D. (1993).
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International Journal for Uncertainty Quantification
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Giles, M.; Duta, M.; Muller, J.; Pierce, N. (2003).
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of our computer simulation, for which we assume the
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Augmented gradient-enhanced kriging (direct method)
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2047:10.1615/Int.J.UncertaintyQuantification.2013006809
1754:On the right we see the reference results for the
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838:{\displaystyle Y\mid x\sim {\mathcal {N}}(Hx,R),}
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2450:Archives of Computational Methods in Engineering
2420:"Performance study of gradient-enhanced Kriging"
2360:AIAA 40th Aerospace Sciences Meeting and Exhibit
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2028:de Baar, J.H.S.; Dwight, R.P.; Bijl, H. (2014).
1699:Example: Drag coefficient of a transonic airfoil
573:LHS-design is available in scripting codes like
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2174:Discrete & Continuous Dynamical Systems - S
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565:When using GEK one takes the following steps:
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2017:
2015:
708:{\displaystyle X\sim {\mathcal {N}}(\mu ,P),}
526:A linear speedup has been demonstrated for a
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1682:Surrogate Modeling Toolbox (SMT) in Python (
1487:Direct GEK (through prior covariance matrix)
206:An adjoint solver allows one to compute the
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1805:2016: Surrogate model construction for two
1793:problem. Demonstration of a linear speedup.
439:. The speedup is the ratio of these costs:
2541:Journal of Chemical Theory and Computation
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2269:PhD Thesis, Delft University of Technology
2012:
1773:GEK has found the following applications:
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1872:ECCOMAS, Vienna, Austria, September 10–14
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1551:, and modify the prior covariance matrix
1121:{\displaystyle K=PH^{T}(R+HPH^{T})^{-1}.}
126:Learn how and when to remove this message
64:Learn how and when to remove this message
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1965:
1735:As an example, consider the flow over a
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1596:, and 3) it is less susceptible to poor
1131:In Kriging, the prior covariance matrix
369:values for the quantity of interest and
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1624:Direct GEK (through observation matrix)
1511:or by modifying the observation matrix
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1818:2020: Molecular geometry optimization.
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2483:Lockwood, B.A.; Anitescu, M. (2012).
570:Create a design of experiment (DoE):
75:
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16:Prediction model used in Engineering
389:partial derivatives in each of the
297:adjoint solves, at a total cost of
190:and US3D. Originally developed for
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898:posterior probability distribution
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14:
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2103:Dwight, R.P.; Han, Z.-H. (2009).
1836:Mitchell, M.; Morris, M. (1992).
1304:where we sum over the dimensions
896:we obtain a normally distributed
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2424:Aerospace Science and Technology
1742:. The airfoil is operating at a
234:scales linearly with the number
80:
44:. Please discuss further on the
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2492:Nuclear Science and Engineering
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1695:location under this framework.
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1048:where we have the gain matrix:
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1344:are the input parameters. The
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662:prior probability distribution
557:in that gradient information.
1:
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1684:https://github.com/SMTorg/SMT
2598:Computational fluid dynamics
176:computational fluid dynamics
7:
2083:10.1016/j.physd.2006.09.017
1807:fluid-structure interaction
1791:fluid-structure interaction
1409:Maximum Likelihood Estimate
868:the observation matrix and
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528:fluid-structure interaction
106:the claims made and adding
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2393:Engineering with Computers
2284:Engineering with Computers
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599:uncertainty quantification
196:uncertainty quantification
2462:10.1007/s11831-017-9226-3
2405:10.1016/j.ast.2012.01.006
2306:10.1007/s00366-018-0590-x
1574:observation uncertainties
1481:observation uncertainties
890:observation uncertainties
139:Gradient-enhanced kriging
2554:10.1021/acs.jctc.0c00257
2263:de Baar, J.H.S. (2014).
1407:can be estimated from a
977:and Kriging covariance:
591:Construct the surrogate:
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1400:{\displaystyle \theta }
1380:{\displaystyle \sigma }
178:(CFD) solvers, such as
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254:of design parameters.
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2593:Mathematical modeling
2196:10.3934/dcdss.2021098
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1679:partial-least squares
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1576:for the gradients in
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42:neutral point of view
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1360:{\displaystyle \mu }
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1337:{\displaystyle \xi }
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731:{\displaystyle \mu }
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432:{\displaystyle N+dN}
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342:{\displaystyle N+dN}
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2118:10.2514/6.2009-2276
2075:2007PhyD..230....1W
1990:2003AIAAJ..41..198G
1941:2006AIAAJ..44.3022D
1906:2015AIAAJ..53.1391D
1668:, see for example.
1600:of the gain matrix
1479:and cannot include
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761:. The observations
609:Predictor equations
530:problem and for a
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313:{\displaystyle 2N}
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91:possibly contains
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2127:978-1-60086-975-4
1935:(12): 3022–3031.
1914:10.2514/1.J053678
1845:Statistica Sinica
1661:{\displaystyle H}
1641:{\displaystyle y}
1613:{\displaystyle K}
1589:{\displaystyle R}
1564:{\displaystyle P}
1544:{\displaystyle y}
1524:{\displaystyle H}
1504:{\displaystyle P}
1472:{\displaystyle h}
1452:{\displaystyle y}
1432:{\displaystyle h}
1317:{\displaystyle k}
1281:
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1144:{\displaystyle P}
892:. After applying
881:{\displaystyle R}
861:{\displaystyle H}
774:{\displaystyle y}
754:{\displaystyle P}
740:covariance matrix
650:{\displaystyle X}
505:
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402:{\displaystyle N}
382:{\displaystyle d}
362:{\displaystyle N}
290:{\displaystyle N}
270:{\displaystyle N}
247:{\displaystyle d}
227:{\displaystyle s}
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93:original research
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37:with its subject.
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1258:
1253:
1247:
1246:
1231:
1230:
1218:
1212:
1209:
1186:
1185:
1173:
1172:
1150:
1148:
1147:
1142:
1127:
1125:
1124:
1119:
1114:
1113:
1101:
1100:
1076:
1075:
1044:
1042:
1041:
1036:
973:
971:
970:
965:
887:
885:
884:
879:
867:
865:
864:
859:
844:
842:
841:
836:
813:
812:
781:have the normal
780:
778:
777:
772:
760:
758:
757:
752:
737:
735:
734:
729:
718:with prior mean
714:
712:
711:
706:
686:
685:
656:
654:
653:
648:
627:Gaussian process
522:
520:
519:
514:
506:
498:
493:
485:
480:
478:
470:
456:
438:
436:
435:
430:
408:
406:
405:
400:
388:
386:
385:
380:
368:
366:
365:
360:
348:
346:
345:
340:
319:
317:
316:
311:
296:
294:
293:
288:
276:
274:
273:
268:
253:
251:
250:
245:
233:
231:
230:
225:
155:response surface
131:
124:
120:
117:
111:
108:inline citations
84:
83:
76:
69:
62:
58:
55:
49:
35:close connection
27:
26:
19:
2613:
2612:
2608:
2607:
2606:
2604:
2603:
2602:
2583:
2582:
2581:
2580:
2533:
2529:
2505:10.1.1.187.6097
2487:
2481:
2477:
2445:
2439:
2435:
2416:
2412:
2389:
2385:
2356:
2352:
2325:
2321:
2280:
2276:
2261:
2257:
2242:10.2514/1.32308
2218:
2211:
2170:
2163:
2151:
2150:
2141:
2140:
2128:
2109:
2101:
2090:
2059:
2055:
2026:
2013:
1970:
1966:
1959:10.2514/1.21744
1950:10.1.1.711.4761
1925:
1921:
1890:
1879:
1867:
1861:
1852:
1840:
1834:
1830:
1825:
1771:
1748:angle of attack
1701:
1692:
1674:
1653:
1650:
1649:
1633:
1630:
1629:
1626:
1605:
1602:
1601:
1581:
1578:
1577:
1556:
1553:
1552:
1536:
1533:
1532:
1516:
1513:
1512:
1496:
1493:
1492:
1489:
1464:
1461:
1460:
1444:
1441:
1440:
1424:
1421:
1420:
1417:
1392:
1389:
1388:
1372:
1369:
1368:
1352:
1349:
1348:
1346:hyperparameters
1329:
1326:
1325:
1309:
1306:
1305:
1274:
1269:
1261:
1254:
1249:
1248:
1239:
1235:
1223:
1219:
1214:
1213:
1211:
1205:
1197:
1193:
1181:
1177:
1165:
1161:
1159:
1156:
1155:
1136:
1133:
1132:
1106:
1102:
1096:
1092:
1071:
1067:
1056:
1053:
1052:
985:
982:
981:
908:
905:
904:
873:
870:
869:
853:
850:
849:
808:
807:
793:
790:
789:
766:
763:
762:
746:
743:
742:
723:
720:
719:
681:
680:
672:
669:
668:
642:
639:
638:
635:
621:to predict the
611:
563:
543:
497:
484:
471:
457:
455:
447:
444:
443:
415:
412:
411:
394:
391:
390:
374:
371:
370:
354:
351:
350:
325:
322:
321:
302:
299:
298:
282:
279:
278:
262:
259:
258:
239:
236:
235:
219:
216:
215:
204:
163:
132:
121:
115:
112:
97:
85:
81:
70:
59:
53:
50:
39:
28:
24:
17:
12:
11:
5:
2611:
2601:
2600:
2595:
2579:
2578:
2527:
2498:(2): 168–195.
2475:
2433:
2410:
2383:
2369:10.1.1.12.4149
2350:
2339:(3): 243–255.
2319:
2274:
2255:
2228:(2): 498–507.
2209:
2161:
2152:|journal=
2126:
2088:
2053:
2040:(3): 205–223.
2011:
1998:10.2514/2.1961
1984:(2): 198–205.
1964:
1919:
1877:
1850:
1827:
1826:
1824:
1821:
1820:
1819:
1816:
1813:
1810:
1803:
1800:
1797:
1794:
1787:
1784:
1781:
1778:
1770:
1767:
1746:of 0.8 and an
1700:
1697:
1691:
1688:
1673:
1670:
1657:
1637:
1625:
1622:
1609:
1585:
1560:
1540:
1520:
1500:
1488:
1485:
1468:
1448:
1428:
1416:
1413:
1396:
1376:
1356:
1333:
1313:
1302:
1301:
1290:
1286:
1277:
1272:
1268:
1264:
1257:
1252:
1245:
1242:
1238:
1234:
1229:
1226:
1222:
1217:
1208:
1204:
1200:
1196:
1192:
1189:
1184:
1180:
1176:
1171:
1168:
1164:
1140:
1129:
1128:
1117:
1112:
1109:
1105:
1099:
1095:
1091:
1088:
1085:
1082:
1079:
1074:
1070:
1066:
1063:
1060:
1046:
1045:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
989:
975:
974:
963:
960:
957:
954:
951:
948:
945:
942:
939:
936:
933:
930:
927:
924:
921:
918:
915:
912:
894:Bayes' Theorem
877:
857:
846:
845:
834:
831:
828:
825:
822:
819:
816:
811:
806:
803:
800:
797:
770:
750:
727:
716:
715:
704:
701:
698:
695:
692:
689:
684:
679:
676:
646:
634:
631:
619:Bayes' Theorem
610:
607:
595:
594:
588:
582:
562:
559:
542:
539:
524:
523:
512:
509:
504:
501:
496:
491:
488:
483:
477:
474:
469:
466:
463:
460:
454:
451:
428:
425:
422:
419:
398:
378:
358:
338:
335:
332:
329:
309:
306:
286:
266:
243:
223:
203:
202:Linear speedup
200:
162:
159:
134:
133:
88:
86:
79:
72:
71:
31:
29:
22:
15:
9:
6:
4:
3:
2:
2610:
2599:
2596:
2594:
2591:
2590:
2588:
2574:
2570:
2565:
2560:
2555:
2550:
2546:
2542:
2538:
2531:
2523:
2519:
2515:
2511:
2506:
2501:
2497:
2493:
2486:
2479:
2471:
2467:
2463:
2459:
2455:
2451:
2444:
2437:
2430:(1): 177–189.
2429:
2425:
2421:
2414:
2406:
2402:
2398:
2394:
2387:
2379:
2375:
2370:
2365:
2362:: 2002–0317.
2361:
2354:
2346:
2342:
2338:
2334:
2333:Technometrics
2330:
2323:
2315:
2311:
2307:
2303:
2298:
2293:
2289:
2285:
2278:
2270:
2266:
2259:
2251:
2247:
2243:
2239:
2235:
2231:
2227:
2223:
2216:
2214:
2205:
2201:
2197:
2193:
2188:
2183:
2179:
2175:
2168:
2166:
2157:
2145:
2137:
2133:
2129:
2123:
2119:
2115:
2108:
2107:
2099:
2097:
2095:
2093:
2084:
2080:
2076:
2072:
2069:(1–2): 1–16.
2068:
2064:
2057:
2048:
2043:
2039:
2035:
2031:
2024:
2022:
2020:
2018:
2016:
2007:
2003:
1999:
1995:
1991:
1987:
1983:
1979:
1975:
1968:
1960:
1956:
1951:
1946:
1942:
1938:
1934:
1930:
1923:
1915:
1911:
1907:
1903:
1899:
1895:
1888:
1886:
1884:
1882:
1873:
1866:
1859:
1857:
1855:
1847:(2): 359–379.
1846:
1839:
1832:
1828:
1817:
1814:
1811:
1808:
1804:
1801:
1798:
1795:
1792:
1788:
1785:
1782:
1779:
1776:
1775:
1774:
1766:
1763:
1759:
1757:
1752:
1749:
1745:
1741:
1738:
1729:
1721:
1713:
1705:
1696:
1687:
1685:
1680:
1669:
1655:
1635:
1621:
1607:
1599:
1583:
1575:
1558:
1538:
1518:
1498:
1484:
1482:
1466:
1446:
1426:
1412:
1410:
1394:
1374:
1354:
1347:
1331:
1311:
1288:
1284:
1275:
1270:
1266:
1262:
1255:
1243:
1240:
1236:
1232:
1227:
1224:
1220:
1206:
1202:
1198:
1194:
1190:
1187:
1182:
1178:
1174:
1169:
1166:
1162:
1154:
1153:
1152:
1138:
1115:
1110:
1107:
1097:
1093:
1089:
1086:
1083:
1080:
1072:
1068:
1064:
1061:
1058:
1051:
1050:
1049:
1032:
1029:
1023:
1020:
1017:
1014:
1008:
1002:
999:
996:
990:
987:
980:
979:
978:
961:
955:
952:
949:
946:
940:
937:
934:
931:
925:
922:
919:
913:
903:
902:
901:
899:
895:
891:
875:
855:
832:
826:
823:
820:
817:
804:
801:
798:
795:
788:
787:
786:
784:
768:
748:
741:
725:
702:
696:
693:
690:
677:
674:
667:
666:
665:
663:
660:
644:
630:
628:
624:
620:
616:
606:
604:
600:
592:
589:
586:
583:
580:
576:
571:
568:
567:
566:
558:
556:
552:
548:
538:
536:
533:
529:
510:
507:
502:
499:
494:
489:
486:
481:
475:
472:
467:
464:
461:
458:
452:
449:
442:
441:
440:
426:
423:
420:
417:
396:
376:
356:
336:
333:
330:
327:
307:
304:
284:
264:
255:
241:
221:
213:
209:
199:
197:
193:
189:
185:
181:
177:
167:
158:
156:
152:
148:
144:
140:
130:
127:
119:
109:
105:
101:
95:
94:
89:This article
87:
78:
77:
68:
65:
57:
47:
43:
38:
36:
30:
21:
20:
2544:
2540:
2530:
2495:
2491:
2478:
2453:
2449:
2436:
2427:
2423:
2413:
2399:(1): 15–34.
2396:
2392:
2386:
2359:
2353:
2336:
2332:
2322:
2287:
2283:
2277:
2268:
2258:
2225:
2222:AIAA Journal
2221:
2177:
2173:
2105:
2066:
2062:
2056:
2037:
2033:
1981:
1978:AIAA Journal
1977:
1967:
1932:
1929:AIAA Journal
1928:
1922:
1897:
1894:AIAA Journal
1893:
1871:
1844:
1831:
1772:
1769:Applications
1764:
1760:
1753:
1734:
1693:
1675:
1627:
1598:conditioning
1490:
1418:
1415:Indirect GEK
1303:
1130:
1047:
976:
847:
717:
636:
629:regression.
612:
603:optimization
596:
590:
584:
569:
564:
544:
525:
256:
205:
192:optimization
173:
161:Introduction
142:
138:
137:
122:
113:
90:
60:
51:
32:
2290:: 157–173.
1744:Mach number
555:uncertainty
2587:Categories
2297:1708.02663
2187:2009.01961
2180:(4): 931.
1823:References
783:likelihood
738:and prior
116:April 2017
100:improve it
54:April 2017
2500:CiteSeerX
2364:CiteSeerX
2271:: 99–101.
2204:221507566
2154:ignored (
2144:cite book
2063:Physica D
1945:CiteSeerX
1809:problems.
1737:transonic
1395:θ
1375:σ
1355:μ
1332:ξ
1267:θ
1237:ξ
1233:−
1221:ξ
1203:∑
1199:−
1191:
1179:σ
1108:−
1018:−
1000:∣
991:
956:μ
950:−
935:μ
923:∣
914:
805:∼
799:∣
726:μ
691:μ
678:∼
532:transonic
151:metamodel
104:verifying
46:talk page
2573:32374164
2522:18465024
2470:54625655
2456:: 1–46.
2250:17895486
2136:59019628
615:Bayesian
601:(UQ) or
561:Approach
551:Bayesian
208:gradient
184:OpenFOAM
2564:7304864
2314:3540630
2230:Bibcode
2071:Bibcode
2006:2106397
1986:Bibcode
1937:Bibcode
1902:Bibcode
1740:airfoil
1411:(MLE).
633:Kriging
623:Kriging
535:airfoil
145:) is a
98:Please
2571:
2561:
2520:
2502:
2468:
2366:
2312:
2248:
2202:
2134:
2124:
2004:
1947:
659:normal
579:Python
575:MATLAB
349:data;
212:linear
180:Fluent
2518:S2CID
2488:(PDF)
2466:S2CID
2446:(PDF)
2310:S2CID
2292:arXiv
2246:S2CID
2200:S2CID
2182:arXiv
2132:S2CID
2110:(PDF)
2002:S2CID
1868:(PDF)
1841:(PDF)
848:with
613:In a
547:noisy
541:Noise
2569:PMID
2156:help
2122:ISBN
1387:and
1324:and
2559:PMC
2549:doi
2510:doi
2496:170
2458:doi
2401:doi
2374:doi
2341:doi
2302:doi
2238:doi
2192:doi
2114:doi
2079:doi
2067:230
2042:doi
1994:doi
1955:doi
1910:doi
1188:exp
988:cov
577:or
188:SU2
143:GEK
102:by
2589::
2567:.
2557:.
2545:16
2543:.
2539:.
2516:.
2508:.
2494:.
2490:.
2464:.
2454:26
2452:.
2448:.
2428:25
2426:.
2422:.
2397:32
2395:.
2372:.
2337:35
2335:.
2331:.
2308:.
2300:.
2288:35
2286:.
2267:.
2244:.
2236:.
2226:46
2224:.
2212:^
2198:.
2190:.
2178:15
2176:.
2164:^
2148::
2146:}}
2142:{{
2130:.
2120:.
2112:.
2091:^
2077:.
2065:.
2036:.
2032:.
2014:^
2000:.
1992:.
1982:41
1980:.
1976:.
1953:.
1943:.
1933:44
1931:.
1908:.
1898:53
1896:.
1880:^
1870:.
1853:^
1843:.
1620:.
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1367:,
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664::
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186:,
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2575:.
2551::
2524:.
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2460::
2407:.
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2304::
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2252:.
2240::
2232::
2206:.
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2138:.
2116::
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2081::
2073::
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2008:.
1996::
1988::
1961:.
1957::
1939::
1916:.
1912::
1904::
1874:.
1656:H
1636:y
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1024:H
1021:K
1015:I
1012:(
1009:=
1006:)
1003:y
997:X
994:(
962:,
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926:y
920:X
917:(
911:E
876:R
856:H
833:,
830:)
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688:(
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508:d
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453:=
450:s
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418:N
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357:N
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334:d
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328:N
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123:(
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
