The details of a method to reduce the computational burden experienced while estimating the optimal model parameters for a Kriging model are presented. A Kriging model is a type of surrogate model that can be used to create a response surface based a set of observations of a computationally expensive system design analysis. This Kriging model can then be used as a computationally efficient surrogate to the original model, providing the opportunity for the rapid exploration of the resulting tradespace. The Kriging model can provide a more complex response surface than the more traditional linear regression response surface through the introduction of a few terms to quantify the spatial correlation of the observations. Implementation details and enhancements to gradient-based methods to estimate the model parameters are presented. It concludes with a comparison of these enhancements to using maximum likelihood estimation to estimate Kriging model parameters and their potential reduction in computational burden. These enhancements include the development of the analytic gradient and Hessian for the log-likelihood equation of a Kriging model that uses a Gaussian spatial correlation function. The suggested algorithm is similar to the SCORING algorithm traditionally used in statistics.

1.
Wang
,
G. G.
, and
Shan
,
S.
, 2007, “
Review of Metamodeling Techniques in Support of Engineering Design Optimization
,”
ASME J. Mech. Des.
0161-8458,
129
, pp.
370
380
.
2.
Qian
,
Z.
,
Seepersad
,
C. C.
,
Joseph
,
V. R.
,
Allen
,
J. K.
, and
Wu
,
C. F. J.
, 2006, “
Building Surrogate Models Based on Detailed and Approximate Simulations
,”
ASME J. Mech. Des.
0161-8458,
128
, pp.
668
677
.
3.
Yang
,
R. J.
,
Wang
,
N.
,
Tho
,
C. H.
,
Bobineau
,
J. P.
, and
Wang
,
B. P.
, 2005, “
Metamodeling Development for Vehicle Frontal Impact Simulation
,”
ASME J. Mech. Des.
1050-0472,
127
(
5
), pp.
1014
1020
.
4.
Pacheco
,
J. E.
,
Amon
,
C. H.
, and
Finger
,
S.
, 2003, “
Bayesian Surrogates Applied to Conceptual Stages of the Engineering Design Process
,”
ASME J. Mech. Des.
1050-0472,
125
(
4
), pp.
664
672
.
5.
Sacks
,
J.
,
Welch
,
W. J.
,
Mitchell
,
T. J.
, and
Wynn
,
H. P.
, 1989, “
Design and Analysis of Computer Experiments
,”
Stat. Sci.
0883-4237,
4
(
4
), pp.
409
435
.
6.
Sasena
,
M. J.
,
Parkinson
,
M.
,
Reed
,
M. P.
,
Papalambros
,
P. Y.
, and
Goovaerts
,
P.
, 2005, “
Improving an Ergonomic Testing Procedure Via Approximation-Based Adaptive Experimental Design
,”
ASME J. Mech. Des.
0161-8458,
127
, pp.
1006
1013
.
7.
Jin
,
R.
,
Chen
,
W.
, and
Sudjianto
,
A.
, 2002, “
On Sequential Sampling for Global Metamodeling in Engineering Design
,” ASME Paper No. DETC2002/DAC-34092.
8.
Jin
,
R.
,
Chen
,
W.
, and
Sudjianto
,
A.
, 2005, “
An Efficient Algorithm for Constructing Optimal Design of Computer Experiments
,”
J. Stat. Plan. Infer.
0378-3758,
134
, pp.
268
287
.
9.
Chen
,
W.
,
Jin
,
R.
, and
Sudjianto
,
A.
, 2005, “
Analytical Variance-Based Global Sensitivity Analysis in Simulation-Based Design Under Uncertainty
,”
ASME J. Mech. Des.
1050-0472,
127
(
5
), pp.
875
886
.
10.
Lophaven
,
S. N.
,
Nielsen
,
B. H.
, and
Sondergaard
,
J.
, 2002, “
DACE—A Matlab Kriging Toolbox, Version 2.0
,” Technical University of Denmark, Report No. IMM-REP-2002-12.
11.
Simpson
,
T. W.
,
Maurey
,
T. M.
,
Korte
,
J. J.
, and
Mistree
,
F.
, 2001, “
Kriging Metamodels for Global Approximation in Simulation-Based Multidisciplinary Design Optimization
,”
AIAA J.
0001-1452,
39
(
12
), pp.
2233
2241
.
12.
Goovaerts
,
P.
, 1997,
Geostatistics for Natural Resources Evaluation
(
Applied Geostatistics Series
),
Oxford University Press
,
New York
.
13.
Welch
,
W. J.
,
Buck
,
R. J.
,
Sacks
,
J.
,
Wynn
,
H. P.
,
Mitchell
,
T. J.
, and
Morris
,
M. D.
, 1992, “
Screening, Predicting, and Computer Experiments
,”
Technometrics
0040-1706,
34
(
1
), pp.
15
25
.
14.
Martin
,
J. D.
, and
Simpson
,
T. W.
, 2005, “
On the Use of Kriging Models to Approximate Deterministic Computer Models
,”
AIAA J.
0001-1452,
43
(
4
), pp.
853
863
.
15.
Joseph
,
V. R.
,
Hung
,
Y.
, and
Sudjianto
,
A.
, 2008, “
Blind Kriging: A New Method for Developing Metamodels
,”
ASME J. Mech. Des.
1050-0472,
130
(
3
), p.
031102
.
16.
Simpson
,
T. W.
,
Poplinski
,
J. D.
,
Koch
,
P. N.
, and
Allen
,
J. K.
, 2001, “
Metamodels for Computer-Based Engineering Design: Survey and Recommendations
,”
Eng. Comput.
0177-0667,
17
(
2
), pp.
129
150
.
17.
Booker
,
A. J.
,
Conn
,
A. R.
,
Dennis
,
J. E.
, Jr.
,
Frank
,
P. D.
,
Trosset
,
M.
, and
Torczon
,
V.
, 1995, “
Global Modeling for Optimization: Boeing/IBM/Rice Collaborative Project 1995 Final Report
,” The Boeing Company, Report No. ISSTECH-95-032.
18.
Currin
,
C.
,
Mitchell
,
T. J.
,
Morris
,
M. D.
, and
Ylvisaker
,
D.
, 1991, “
Bayesian Prediction of Deterministic Functions, With Applications to the Design and Analysis of Computer Experiments
,”
J. Am. Stat. Assoc.
0162-1459,
86
(
416
), pp.
953
963
.
19.
Mardia
,
K.
, and
Marshall
,
R.
, 1984, “
Maximum Likelihood Estimation of Models for Residual Covariance in Spatial Regression
,”
Biometrika
0006-3444,
71
(
1
), pp.
135
146
.
20.
Kitanidis
,
P. K.
, 1986, “
Parameter Uncertainty in Estimation of Spatial Functions: Bayesian Analysis
,”
Water Resour. Res.
0043-1397,
22
, pp.
499
507
.
21.
Mardia
,
K.
, and
Watkins
,
A. J.
, 1989, “
On Multimodality of the Likelihood in the Spatial Linear Model
,”
Biometrika
0006-3444,
76
(
2
), pp.
289
295
.
22.
Warnes
,
J. J.
, and
Ripley
,
B. D.
, 1987, “
Problems With Likelihood Estimation of Covariance Function of Spatial Gaussian Processes
,”
Biometrika
0006-3444,
74
(
3
), pp.
640
642
.
23.
Sacks
,
J.
,
Schiller
,
S. B.
, and
Welch
,
W. J.
, 1989, “
Design for Computer Experiments
,”
Technometrics
0040-1706,
31
(
1
), pp.
41
47
.
24.
Osio
,
I. G.
, and
Amon
,
C. H.
, 1996, “
An Engineering Design Methodology With Multistage Bayesian Surrogate and Optimal Sampling
,”
Res. Eng. Des.
0934-9839,
8
(
4
), pp.
189
206
.
25.
Booker
,
A. J.
,
Dennis
,
J. E.
, Jr.
,
Frank
,
P. D.
,
Serafini
,
D. B.
,
Torczon
,
V.
, and
Trosset
,
M.
, 1999, “
A Rigorous Framework for Optimization of Expensive Functions by Surrogates
,”
Struct. Optim.
0934-4373,
17
(
1
), pp.
1
13
.
26.
Efron
,
B.
, and
Hinkley
,
D.
, 1978, “
Assessing the Accuracy of the Maximum Likelihood Estimator: Observed Versus Expected Fisher Information
,”
Biometrika
0006-3444,
65
(
3
), pp.
457
482
.
27.
Marquardt
,
D. W.
, 1963, “
An Algorithm for Least-Squares Estimation of Nonlinear Parameters
,”
J. Soc. Ind. Appl. Math.
0368-4245,
11
(
2
), pp.
431
441
.
28.
Hemmerle
,
W. J.
, and
Hartley
,
H. O.
, 1973, “
Computing Maximum Likelihood Estimates for the Mixed A. O. V. Model Using the W Transform
,”
Technometrics
0040-1706,
15
(
4
), pp.
819
831
.
29.
Jennrich
,
R. I.
, and
Sampson
,
P. F.
, 1976, “
Newton–Raphson and Related Algorithms for Maximum Likelihood Variance Component Estimation
,”
Technometrics
0040-1706,
18
(
1
), pp.
11
17
.
30.
Martin
,
J. D.
, and
Simpson
,
T. W.
, 2006, “
A Methodology to Manage Uncertainty During System-Level Conceptual Design
,”
ASME J. Mech. Des.
0161-8458,
128
(
4
), pp.
959
968
.
31.
Searle
,
S. R.
,
Casella
,
G.
, and
McCulloch
,
C. E.
, 1992,
Variance Components
,
Wiley
,
New York
.
32.
Martin
,
J. D.
, and
Simpson
,
T. W.
, 2002, “
Use of Adaptive Metamodeling for Design Optimization
,” AIAA Paper No. AIAA-2002-5631.
33.
Martin
,
J. D.
, and
Simpson
,
T. W.
, 2004, “
A Monte Carlo Simulation of the Kriging Model
,” AIAA Paper No. AIAA-2004-4483.
34.
White
,
H.
, 1982, “
Maximum Likelihood Estimation of Misspecified Models
,”
Econometrica
0012-9682,
50
(
1
), pp.
1
26
.
You do not currently have access to this content.