Technical Briefs

Numerical Strategies to Reduce the Effect of Ill-Conditioned Correlation Matrices and Underflow Errors in Kriging

[+] Author and Article Information
Lukas J. Haarhoff

Powertech Transformers,
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, South Africa
e-mail: skok@csir.co.za

Daniel N. Wilke

Department of Mechanical Engineering,
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 135(4), 044502 (Mar 26, 2013) (4 pages) Paper No: MD-11-1442; doi: 10.1115/1.4023631 History: Received October 31, 2011; Revised February 06, 2013

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.

Copyright © 2013 by ASME
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Martin, J. D., and Simpson, T. W., 2005, “Use of Kriging Models to Approximate Deterministic Computer Models,” AIAA J., 43(4), pp. 853–863. [CrossRef]
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, 34(1), pp. 15–25. [CrossRef]
Booker, A. J., Dennis, J. E., Frank, P. D., Serafini, D. B., Torczon, V., and Trosset, M. W., 1999, “A Rigorous Framework for Optimization of Expensive Functions by Surrogates,” Struct. Multidiscip. Optim., 17, pp. 1–13. [CrossRef]
Martin, J. D., 2009, “Computational Improvements to Estimating Kriging Metamodel Parameters,” ASME J. Mech. Design, 131, 084501. [CrossRef]
Yamazaki, W., Rumpfkeil, M. P., and Mavriplis, D. J., 2010, “Design Optimization Utilizing Gradient/Hessian Enhanced Surrogate Model,” AIAA Paper No. 2010-4363.
Ranjan, P., Haynes, R., and Karsten, R., 2010, “Gaussian Process Models and Interpolators for Deterministic Computer Simulators,” e-print arXiv:cond.mat/1003.1315v2.
Dwight, R. P., and Han, Z., 2009, “Efficient Uncertainty Quantification Using Gradient-Enhanced Kriging,” AIAA Paper No. 2009-2276.
Laurenceau, J., and Sagaut, P., 2008, “Building Efficient Response Surfaces of Aerodynamic Functions With Kriging and Cokriging,” AIAA J., 46(2), pp. 498–507. [CrossRef]
Simpson, T. W., 1998, “A Concept Exploration Method for Product Family Design,” Ph.D. thesis, Georgia Institute of Technology, Atlanta.
MATLAB 7.14, 2012, The Mathworks Inc, Natick, MA.
Gentle, J. E., 2007, Matrix Algebra: Theory, Computations, and Applications in Statistics, Springer, New York.
Ciarlet, P. G., 1989, Introduction to Numerical Linear Algebra and Optimization, Cambridge University, Cambridge.
Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., and Sorensen, D., 1999, LAPACK Users guide, 3rd ed., Society for Industrial and Applied Mathematics, Philadelphia.
IEEE, 2008, IEEE Standard 754-2008: IEEE Standard for Floating-Point Arithmetic, IEEE, New York.
Rust, B. W., 1994, “Perturbation Bounds for Linear Regression Problems,” Comput. Sci. Stat., 26, pp. 965–984.
Johansson, F. 2011, “mpmath: A Python Library for Arbitrary-Precision Floating-Point Arithmetic (version 0.14),” http://code.google.com/p/mpmath/
Hock, W., and Schittkowski, K., 1981, “Test Examples for Nonlinear Programming Code,” Lecture Notes in Economical and Mathematical Systems, Vol. 187, Springer-Verlag, Berlin Heidelberg New York.
Johannson, F., and the MPMATH Development Team, 2010, Version 0.14.


Grahic Jump Location
Fig. 1

Computing the Kriging interpolation errors using either an explicit inverse for R or by using decomposition and back substitution for test problem

Grahic Jump Location
Fig. 2

Using R1 and decomposition and back substitution for Hock 2, DOE 2

Grahic Jump Location
Fig. 3

Using R1 and decomposition and back substitution for Hock 10, DOE 1

Grahic Jump Location
Fig. 4

Using R1 and decomposition and back substitution for Hock 10, DOE 2

Grahic Jump Location
Fig. 5

Using R−1 and decomposition and back substitution for Hock 26, DOE 3

Grahic Jump Location
Fig. 6

Comparing the numerical computation of ln|R| following the multiplicative computation given by Eq. (13), against the additive computation given by Eq. (14), with arbitrary precision results provided as a reference



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