An adaptive sampling method for Kriging surrogate model with multiple outputs
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ORIGINAL ARTICLE
An adaptive sampling method for Kriging surrogate model with multiple outputs Zhangming Zhai1,2 · Haiyang Li1 · Xugang Wang2 Received: 2 June 2020 / Accepted: 11 August 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract The sample distribution has a vital influence on the quality of a Kriging surrogate model, which may further influence the required cost or convergence of the surrogate model-based design and optimization problems. Adaptive sampling methods utilize the information from existing samples to reasonably allocate the sequential samples, which can generally build a more accurate Kriging surrogate model under the same computational budget. However, most of the existing adaptive sampling methods for the Kriging surrogate model are only available for single-output problems, and there are few studies on problems with multiple responses. In this paper, an adaptive sampling method based on Delaunay triangulation and technique for order preference by similarity to ideal solution (TOPSIS) is proposed for Kriging surrogate model with multiple outputs (mKMDT). In the proposed mKMDT, Delaunay triangulation is used to partition the design space into multiple triangle regions, whose area denotes the dispersion of the sample points. The prediction error at each triangle’s centroid represents the local approximation error. Specifically, three different strategies are developed when allocating weights to the area and the prediction error of each triangle with the entropy method and the TOPSIS method. The performance of the proposed method is illustrated through numerical examples with different numbers of outputs and a collision problem between the missile and the adapter. Results show that the proposed method can construct an accuracy surrogate model with few samples, which is useful for practical engineering design problems with multiple outputs. Keywords Adaptive sampling method · Multiple outputs · Kriging surrogate model · Simulation-based design
1 Introduction In engineering design and optimization problems, surrogate models have been wildly used to replace computational expensive simulation models to reduce the computational cost [1–7]. Commonly used surrogate models includes polynomial response surface (PRS) model [8], Kriging model [9–11], radial basis function (RBF) model [12, 13], and support vector regression (SVR) model [14]. Among these surrogate models, the Kriging model can not only predict the responses of the unobserved points, but also provide the predicted variance. In consequence, it has gained widespread applications in many fields, e.g., aerospace [15, 16], * Zhangming Zhai [email protected] 1
National University of Defense Technology, Changsha 410073, China
Beijing Institute of Space System Engineering, Beijing 100076, China
2
automobile [17], and ocean engineering [18]. The quality of a surrogate model has a great influence on the computational cost or the convergence of the surrogate model-based design optimization problems. Under the limite
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