An adaptive PCE-HDMR metamodeling approach for high-dimensional problems
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 pre...
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| Published in: | Structural and multidisciplinary optimization Vol. 64; no. 1; pp. 141 - 162 |
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| Main Authors: | , , , , |
| Format: | Journal Article |
| Language: | English |
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Berlin/Heidelberg
Springer Berlin Heidelberg
01.07.2021
Springer Nature B.V |
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| ISSN: | 1615-147X, 1615-1488 |
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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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| AbstractList | 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. |
| Author | Luo, Min Duan, Libin Zhang, Jian Gong, Weijie Yue, Xinxin |
| Author_xml | – sequence: 1 givenname: Xinxin surname: Yue fullname: Yue, Xinxin organization: Department of Mechanics and Engineering Science, Jiangsu University – sequence: 2 givenname: Jian orcidid: 0000-0002-7497-0646 surname: Zhang fullname: Zhang, Jian email: jianzhang@u.nus.edu organization: Department of Mechanics and Engineering Science, Jiangsu University – sequence: 3 givenname: Weijie surname: Gong fullname: Gong, Weijie organization: Department of Mechanics and Engineering Science, Jiangsu University – sequence: 4 givenname: Min surname: Luo fullname: Luo, Min organization: Ocean College, Zhejiang University – sequence: 5 givenname: Libin surname: Duan fullname: Duan, Libin organization: School of Automotive and Traffic Engineering, Jiangsu University |
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| Keywords | Polynomial chaos expansion (PCE) Adaptive sampling High-dimensional model representation (HDMR) Metamodeling Design optimization |
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| SubjectTerms | Algorithms Approximation Computational Mathematics and Numerical Analysis Design engineering Design optimization Dimensional analysis Engineering Engineering Design Functions (mathematics) Mathematical analysis Mathematical functions Metamodels Polynomials Rectangles Research Paper Robustness (mathematics) Sampling methods Structural hierarchy Theoretical and Applied Mechanics |
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