Structural and Multidisciplinary Optimization

, Volume 59, Issue 1, pp 21–42 | Cite as

Comparative study of HDMRs and other popular metamodeling techniques for high dimensional problems

  • Liming Chen
  • Hu WangEmail author
  • Fan Ye
  • Wei Hu


The efficiency of optimization for the high dimensional problem has been improved by the metamodeling techniques in multidisciplinary in the past decades. In this study, comparative studies are implemented for high dimensional problems on the accuracy of four popular metamodeling methods, Kriging (KRG), radial basis function (RBF), least square support vector regression (LSSVR) and cut-high dimensional model representation (cut-HDMR) methods. Besides, HDMR methods with different basis functions are considered, including KRG-HDMR, RBF-HDMR and SVR-HDMR. Four factors that might influence the quality of metamodeling methods involving parameter interaction of problems, sample sizes, noise level and sampling strategies are considered. The results show that the LSSVR with Gaussian kernel, using Latin hypercube sampling (LHS) strategy, constructs more accurate metamodels than the KRG. The RBF with Gaussian basis function performs poor in the group. Generally, cut-HDMR methods perform much better than the other metamodeling methods when handling the function with weak parameter interaction, but not better when handling the function with strong parameter interaction.


Metamodel High dimensional problem HDMR Parameter interaction 



Correlation function


Sample point


Correlation coefficient


Vector of sample point

\( \widehat{y} \)

Prediction of sample point


Weigh coefficient


Predefined polynomial basis function


Realization of stochastic process




Basis functions


Cost function


A vector of weights


Penalty parameter


Error variable


Evaluation of sample point


A set of sample point


Nonlinear mapping


Model offset


Lagrange multiplier


Kernel function


Tuning parameter


Design domain


Vector consisting linear variable terms

E, F



Standard Gaussian distribution


Random number sampled


Center points


Side length


Unit vector


Best function values



Point index




Point index


Number of sample point



This work has been supported by National Key Research and Development Program of China 2017YFB0203701 and Project of the Program of National Natural Science Foundation of China under the Grant Numbers 11572120.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018
corrected publication September/2018

Authors and Affiliations

  1. 1.State Key Laboratory of Advanced Design and Manufacturing for Vehicle BodyHunan UniversityChangshaChina
  2. 2.Joint Center for Intelligent New Energy VehicleShanghaiChina

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