Kriging is used extensively as a metamodel in multidisciplinary design optimization. The correlation matrix used in Kriging metamodeling frequently becomes ill-conditioned. Therefore different numerical methods used to solve the Kriging equations affect the search for the optimum Kriging parameters and the ability of the Kriging surface to accurately interpolate known data points. We illustrate this by firstly computing the inverse of the correlation matrix in the Kriging equations, and secondly by solving the systems of equations using decomposition and back substitution, thereby avoiding the inversion of the correlation matrix. Our results clearly show that by decomposing and back substituting, the interpolation accuracy is maintained for significantly higher condition numbers. We then show that computing the natural logarithm of the determinant using additive calculations as opposed to multiplicative calculations significantly reduces numerical underflow errors encountered when searching for the optimum Kriging parameters. Although the effect of decomposition and back substitution are known, and the underflow difficulties when computing the natural logarithm of the determinant of the correlation matrix has been mentioned in passing in Kriging literature, this work clearly quantifies and reinforces these methods, hopefully for the benefit of researchers entering the field.
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April 2013
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Numerical Strategies to Reduce the Effect of Ill-Conditioned Correlation Matrices and Underflow Errors in Kriging
Lukas J. Haarhoff,
Lukas J. Haarhoff
Powertech Transformers,
P.O. Box 691,
Pretoria, Gauteng, 0001,
e-mail: johan.haarhoff@pttransformers.co.za
P.O. Box 691,
Pretoria, Gauteng, 0001,
South Africa
e-mail: johan.haarhoff@pttransformers.co.za
Search for other works by this author on:
Schalk Kok,
Schalk Kok
Advanced Mathematical Modelling,
CSIR Modelling and Digital Science,
P.O. Box 395,
Pretoria, Gauteng, 0001,
e-mail: skok@csir.co.za
CSIR Modelling and Digital Science,
P.O. Box 395,
Pretoria, Gauteng, 0001,
South Africa
e-mail: skok@csir.co.za
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Daniel N. Wilke
Daniel N. Wilke
Department of Mechanical Engineering,
e-mail: nico.wilke@up.ac.za
University of Pretoria
,Private bag X20
,Hatfield, Pretoria, Gauteng, 0028
, South Africa
e-mail: nico.wilke@up.ac.za
Search for other works by this author on:
Lukas J. Haarhoff
Powertech Transformers,
P.O. Box 691,
Pretoria, Gauteng, 0001,
e-mail: johan.haarhoff@pttransformers.co.za
P.O. Box 691,
Pretoria, Gauteng, 0001,
South Africa
e-mail: johan.haarhoff@pttransformers.co.za
Schalk Kok
Advanced Mathematical Modelling,
CSIR Modelling and Digital Science,
P.O. Box 395,
Pretoria, Gauteng, 0001,
e-mail: skok@csir.co.za
CSIR Modelling and Digital Science,
P.O. Box 395,
Pretoria, Gauteng, 0001,
South Africa
e-mail: skok@csir.co.za
Daniel N. Wilke
Department of Mechanical Engineering,
e-mail: nico.wilke@up.ac.za
University of Pretoria
,Private bag X20
,Hatfield, Pretoria, Gauteng, 0028
, South Africa
e-mail: nico.wilke@up.ac.za
Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received October 31, 2011; final manuscript received February 6, 2013; published online March 26, 2013. Assoc. Editor: Timothy W. Simpson.
J. Mech. Des. Apr 2013, 135(4): 044502 (4 pages)
Published Online: March 26, 2013
Article history
Received:
October 31, 2011
Revision Received:
February 6, 2013
Citation
Haarhoff, L. J., Kok, S., and Wilke, D. N. (March 26, 2013). "Numerical Strategies to Reduce the Effect of Ill-Conditioned Correlation Matrices and Underflow Errors in Kriging." ASME. J. Mech. Des. April 2013; 135(4): 044502. https://doi.org/10.1115/1.4023631
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