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An adaptive PCE-HDMR metamodeling approach for high-dimensional problems

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Abstract

Metamodel-based high-dimensional model representation (HDMR) has recently been developed as a promising tool for approximating high-dimensional and computationally expensive problems in engineering design and optimization. However, current stand-alone Cut-HDMRs usually come across the problem of prediction uncertainty while combining an ensemble of metamodels with Cut-HDMR results in an implicit and inefficient process in response approximation. To this end, a novel stand-alone Cut-HDMR is proposed in this article by taking advantage of the explicit polynomial chaos expansion (PCE) and hierarchical Cut-HDMR (named PCE-HDMR). An intelligent dividing rectangles (DIRECT) sampling method is adopted to adaptively refine the model. The novelty of the PCE-HDMR is that the proposed multi-hierarchical algorithm structure by integrating PCE with Cut-HDMR can efficiently and robustly provide simple and explicit approximations for a wide class of high-dimensional problems. An analytical function is first used to illustrate the modeling principles and procedures of the algorithm, and a comprehensive comparison between the proposed PCE-HDMR and other well-established Cut-HDMRs is then made on fourteen representative mathematical functions and five engineering examples with a wide scope of dimensionalities. The results show that the proposed PCE-HDMR has much superior accuracy and robustness in terms of both global and local error metrics while requiring fewer number of samples, and its superiority becomes more significant for polynomial-like functions, higher-dimensional problems, and relatively larger PCE degrees.

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Funding

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11872190, 51805221), Six Talent Peaks Project in Jiangsu Province (Grant No. 2017-KTHY-010), and Research Start-up Foundation for Jinshan Distinguished Professor at Jiangsu University (Grant No. 4111480003).

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Authors and Affiliations

  1. Department of Mechanics and Engineering Science, Jiangsu University, Zhenjiang, 212013, China

    Xinxin Yue, Jian Zhang & Weijie Gong

  2. Ocean College, Zhejiang University, Zhoushan, 316021, China

    Min Luo

  3. School of Automotive and Traffic Engineering, Jiangsu University, Zhenjiang, 212013, China

    Libin Duan

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  1. Xinxin Yue
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  2. Jian Zhang
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  3. Weijie Gong
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  4. Min Luo
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Correspondence to Jian Zhang.

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Replication of results

The original codes for the illustrative example in Section 3.4 and fourteen benchmark functions and five engineering examples in Section 4 are available in the supplementary materials, i.e., IllustrativeExample.m, F1.m ~ F15.m, and E1.m ~ E5.m.

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Yue, X., Zhang, J., Gong, W. et al. An adaptive PCE-HDMR metamodeling approach for high-dimensional problems. Struct Multidisc Optim 64, 141–162 (2021). https://doi.org/10.1007/s00158-021-02866-7

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  • DOI: https://doi.org/10.1007/s00158-021-02866-7

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