On the maximum likelihood training of gradient-enhanced spatial Gaussian processes

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Spatial Gaussian processes, alias spatial linear models or Kriging estimators, are a
powerful and well-established tool for the design and analysis of computer experiments in a multitude
of engineering applications. A key challenge in constructing spatial Gaussian processes is the training
of the predictor by numerically optimizing its associated maximum likelihood function depending on
so-called hyper-parameters. This is well understood for standard Kriging predictors, i.e., without
considering derivative information. For gradient-enhanced Kriging predictors it is an open question
of whether to incorporate the cross-correlations between the function values and their partial deriva-
tives in the maximum likelihood estimation. In this paper it is proved that in consistency with the
model assumptions, both the autocorrelations and the aforementioned cross-correlations must be
considered when optimizing the gradient-enhanced predictor’s likelihood function. The proof works
by computational rather than probabilistic arguments and exposes as a secondary effect the connec-
tion between the direct and the indirect approach to gradient-enhanced Kriging, both of which are
widely used in applications. The theoretical findings are illustrated on an academic example as well
as on an aerodynamic engineering application.
Original languageEnglish
JournalSIAM Journal on Scientific Computing
Issue number6
Pages (from-to)A2554–A2574
Number of pages21
Publication statusPublished - 2013
Externally publishedYes


  • design and analysis of computer experiments
  • gradient-enhanced Kriging
  • Gaussian process
  • maximum likelihood
  • Design and analysis of computer experiments
  • Surrogate model
  • Hyper-parameter training
  • Gradient-enhanced Kriging
  • Response surface
  • Spatial linear model
  • Maximum likelihood


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