“…For clarity, Figure 2 shows a schematic of the bisection sampling process. Note that other advanced sequential sampling methods 18,19 apart from the bisection sampling method can also be applied in this step, which is however beyond the scope of the present work. Figure 2.A schematic of the bisection sampling method.…”
“…17 and Li et al. 18 adopted the dividing rectangles (DIRECT) method and a projection method respectively to generate new samples for metamodel-assisted HDMR construction. However, no convincing evidence was provided for the effectiveness of these methods.…”
A major challenge of metamodeling in simulation-based engineering design optimization is to handle the “curse of dimensionality,” i.e. the exponential growth of computational cost with increase of problem dimensionality. Encouragingly, it has been reported recently that a high-dimensional model representation assisted by a radial basis function is capable of deriving high-dimensional input–output relationships at dramatically reduced computational cost. In this article, support vector regression is employed as an alternative to be coupled with high-dimensional model representation for the metamodeling of high-dimensional problems. In particular, the bisection sampling method is proposed to be used in the metamodeling process to generate high-quality training samples. Testing and comparison results show that the developed bisection-sampling-based support vector regression–high-dimensional model representation metamodeling technique can achieve high modeling accuracy with a smaller number of training sample evaluations. For the problem examined in this study, the bisection-sampling-based support vector regression–high-dimensional model representation enables high modeling accuracy and linear computational complexity as the problem dimensionality increases. Analysis of this performance advantage shows that the use of bisection method enables the developed metamodeling technique to be more effective in dealing with high-dimensional problems.
“…For clarity, Figure 2 shows a schematic of the bisection sampling process. Note that other advanced sequential sampling methods 18,19 apart from the bisection sampling method can also be applied in this step, which is however beyond the scope of the present work. Figure 2.A schematic of the bisection sampling method.…”
“…17 and Li et al. 18 adopted the dividing rectangles (DIRECT) method and a projection method respectively to generate new samples for metamodel-assisted HDMR construction. However, no convincing evidence was provided for the effectiveness of these methods.…”
A major challenge of metamodeling in simulation-based engineering design optimization is to handle the “curse of dimensionality,” i.e. the exponential growth of computational cost with increase of problem dimensionality. Encouragingly, it has been reported recently that a high-dimensional model representation assisted by a radial basis function is capable of deriving high-dimensional input–output relationships at dramatically reduced computational cost. In this article, support vector regression is employed as an alternative to be coupled with high-dimensional model representation for the metamodeling of high-dimensional problems. In particular, the bisection sampling method is proposed to be used in the metamodeling process to generate high-quality training samples. Testing and comparison results show that the developed bisection-sampling-based support vector regression–high-dimensional model representation metamodeling technique can achieve high modeling accuracy with a smaller number of training sample evaluations. For the problem examined in this study, the bisection-sampling-based support vector regression–high-dimensional model representation enables high modeling accuracy and linear computational complexity as the problem dimensionality increases. Analysis of this performance advantage shows that the use of bisection method enables the developed metamodeling technique to be more effective in dealing with high-dimensional problems.
“…Building a metamodel involves two procedures: (1) employing design of experiments (DOEs) to sample the computer simulation; and (2) selecting an approximation model to represent the data and fit the model with the sampling data . Various metamodels have been developed, such as the polynomial model (Montgomery 2007), kriging model (Chen et al 2014;Li et al 2013), radial basis functions (RBF) (Fang and Horstemeyer 2006), multivariate adaptive regression splines (MARS) (Friedman 1991), support vector regression (Clarke, Griebsch, and Simpson 2005;, high-dimensional model representation (Shan and Wang 2010;Wang, Tang, and Li 2011;Hajikolaei and Wang 2012;Li, Wang, and Li 2012), and multisurrogate models (Zerpa et al 2005;Goel et al 2006;Zhang, Chowdhury, and Messac 2012;Zheng et al 2013), which combine some of the basic metamodels. *Corresponding author.…”
The accuracy of metamodelling is determined by both the sampling and approximation. This article proposes a new sampling method based on the zeros of Chebyshev polynomials to capture the sampling information effectively. First, the zeros of one-dimensional Chebyshev polynomials are applied to construct Chebyshev tensor product (CTP) sampling, and the CTP is then used to construct high-order multi-dimensional metamodels using the 'hypercube' polynomials. Secondly, the CTP sampling is further enhanced to develop Chebyshev collocation method (CCM) sampling, to construct the 'simplex' polynomials. The samples of CCM are randomly and directly chosen from the CTP samples. Two widely studied sampling methods, namely the Smolyak sparse grid and Hammersley, are used to demonstrate the effectiveness of the proposed sampling method. Several numerical examples are utilized to validate the approximation accuracy of the proposed metamodel under different dimensions.
“…From this point of view, an immediate idea is that if we find out the reproducing kernel K l (x, y) of the space S l defined by Eq. (4), every function g in S l will have a natural representation, that is, (11) here {χ i } is the input sample set such that g can be uniquely determined by {χ i , g(χ i )}, and the reproducing kernel K l (x, y) is independent of the given function g(x). Usually, we consider the approximate form like the situation of RBF interpolatioñ…”
Section: Introductionmentioning
confidence: 99%
“…However, high-dimensional model representations (HDMR) [1,2] opens a remarkable opportunity to overcome the curse of dimensionality. By introducing an ansatz that for most physical systems, only relatively low order correlations of the input variables will have an impact on the output [2,9,10], a high-dimensional function used to describe a well-defined physical system can be approximatively viewed as a sum of low-dimensional functions [11][12][13][14][15].…”
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