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Efficient high-dimensional metamodeling strategy using selectively high-ordered kriging HDMR (SH-K-HDMR)

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Abstract

Selectively high-ordered kriging high dimension model representation (SH-K-HDMR) is proposed for efficient metamodeling of high-dimensional, expensive, blackbox (HEB) problems. SH-K-HDMR is a kind of cut HDMR, which selectively models high order interaction components. SH-K-HDMR starts with two groups of samples, which are for HDMR and model accuracy assessment, respectively. Variance of kriging prediction error is used for sequential sampling, and an interaction measure Hij… is suggested and used to choose high order interaction components of cut HDMR to be modeled. The group of samples for model accuracy assessment is also used for surrogate modeling at the end of the metamodeling process, to eliminate waste of samples. Performance of the SH-K-HDMR is compared to conventional metamodeling methods, namely full-dimension kriging and kriging-HDMR. The comparison study shows that SH-K-HDMR is superior when applied to high-dimensional and nonlinear engineering problems with strong interactions between variables.

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Abbreviations

b, w :

Trend function weights and kriging weights

d :

Number of relevant variables

f :

Basis functions of trend \(\mu(\overrightarrow{x})\)

g :

Component functions of HDMR

H :

Measure of interaction

i, j :

Indices

N, m :

Number of samples and trend basis functions

n ():

Number of set elements

\(\overrightarrow{x}\) :

Design variables

\(y(\overrightarrow{x}),\hat{y}(\overrightarrow{x})\) :

Target variable and its surrogate model

\(Z(\overrightarrow{x})\) :

Random part of the target variable

\(y(\Delta\overrightarrow{x})\) :

Variogram of \(Z(\overrightarrow{x})\)

ε target :

Target accuracy

λ :

Lagrange multipliers

\(\mu(\overrightarrow{x})\) :

Trend part of target variable

σ 2p :

Variance of kriging prediction error

RMSE:

Root mean square error

NRMSE:

Normalized root mean square error

NME:

Normalized maximum error

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Author information

Authors and Affiliations

  1. Aerospace Technology Research Institute, Agency for Defense Development, Daejeon, 34186, Korea

    Donghyun Kim

  2. Department of Mechanical Engineering, KAIST, Daejeon, 34141, Korea

    Ikjin Lee

Authors
  1. Donghyun Kim
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  2. Ikjin Lee
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Corresponding author

Correspondence to Ikjin Lee.

Additional information

Donghyun Kim received B.S. degree in Mechanical Engineering from KAIST. He is currently working in ADD, serving for his mandatory military service. His research interests include fluid mechanics, control engineering, and their applications in fluid machineries.

Ikjin Lee is an Associate Professor of Mechanical Engineering at KAIST. He received his Ph.D. in Mechanical Engineering at the University of Iowa. He joined KAIST in 2013. His research interests include system reliability analysis, design optimization, surrogate modeling, and system robustness analysis and design.

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Kim, D., Lee, I. Efficient high-dimensional metamodeling strategy using selectively high-ordered kriging HDMR (SH-K-HDMR). J Mech Sci Technol 35, 5099–5105 (2021). https://doi.org/10.1007/s12206-021-1026-x

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  • DOI: https://doi.org/10.1007/s12206-021-1026-x

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