### Resumé

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.

Originalsprog | Engelsk |
---|---|

Tidsskrift | S I A M Journal on Scientific Computing |

Vol/bind | 35 |

Udgave nummer | 6 |

Sider (fra-til) | A2554–A2574 |

Antal sider | 21 |

ISSN | 1064-8275 |

Status | Udgivet - 2013 |

Udgivet eksternt | Ja |

### Fingeraftryk

### Citer dette

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*S I A M Journal on Scientific Computing*, bind 35, nr. 6, s. A2554–A2574.

**On the maximum likelihood training of gradient-enhanced spatial Gaussian processes.** / Zimmermann, Ralf.

Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › peer review

TY - JOUR

T1 - On the maximum likelihood training of gradient-enhanced spatial Gaussian processes

AU - Zimmermann, Ralf

PY - 2013

Y1 - 2013

N2 - Spatial Gaussian processes, alias spatial linear models or Kriging estimators, are apowerful and well-established tool for the design and analysis of computer experiments in a multitudeof engineering applications. A key challenge in constructing spatial Gaussian processes is the trainingof the predictor by numerically optimizing its associated maximum likelihood function depending onso-called hyper-parameters. This is well understood for standard Kriging predictors, i.e., withoutconsidering derivative information. For gradient-enhanced Kriging predictors it is an open questionof 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 themodel assumptions, both the autocorrelations and the aforementioned cross-correlations must beconsidered when optimizing the gradient-enhanced predictor’s likelihood function. The proof worksby 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 arewidely used in applications. The theoretical findings are illustrated on an academic example as wellas on an aerodynamic engineering application.

AB - Spatial Gaussian processes, alias spatial linear models or Kriging estimators, are apowerful and well-established tool for the design and analysis of computer experiments in a multitudeof engineering applications. A key challenge in constructing spatial Gaussian processes is the trainingof the predictor by numerically optimizing its associated maximum likelihood function depending onso-called hyper-parameters. This is well understood for standard Kriging predictors, i.e., withoutconsidering derivative information. For gradient-enhanced Kriging predictors it is an open questionof 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 themodel assumptions, both the autocorrelations and the aforementioned cross-correlations must beconsidered when optimizing the gradient-enhanced predictor’s likelihood function. The proof worksby 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 arewidely used in applications. The theoretical findings are illustrated on an academic example as wellas on an aerodynamic engineering application.

KW - design and analysis of computer experiments

KW - gradient-enhanced Kriging

KW - Gaussian process

KW - maximum likelihood

M3 - Journal article

VL - 35

SP - A2554–A2574

JO - S I A M Journal on Scientific Computing

JF - S I A M Journal on Scientific Computing

SN - 1064-8275

IS - 6

ER -