In this work, a cut high-dimensional model representation (cut-HDMR) expansion based on multiple anchors is constructed via the clustering method. Specifically, a set of random input realizations is drawn from the parameter space and grouped by the centroidal Voronoi tessellation (CVT) method. Then for each cluster, the centroid is set as the reference, thereby the corresponding zeroth-order term can be determined directly. While for non-zero order terms of each cut-HDMR, a set of discrete points is selected for each input component, and the Lagrange interpolation method is applied. For a new input, the cut-HDMR corresponding to the nearest centroid is used to compute its response. Numerical experiments with high-dimensional integral and elliptic stochastic partial differential equation as backgrounds show that the CVT based multiple anchors cut-HDMR can alleviate the negative impact of a single inappropriate anchor point, and has higher accuracy than the average of several expansions.

翻译：本文通过聚类方法构建了一种基于多锚点的截断高维模型表示（cut-HDMR）展开。具体而言，从参数空间中抽取一组随机输入样本，并使用质心沃罗诺伊图（CVT）方法进行分组。随后，将每个簇的质心设为参考点，从而直接确定对应的零阶项。对于每个截断HDRM的非零阶项，为每个输入分量选择一组离散点，并应用拉格朗日插值方法。对于新的输入，使用最近质心对应的截断HDMR计算其响应。以高维积分和椭圆型随机偏微分方程为背景的数值实验表明，基于CVT的多锚点截断HDMR能够减轻单一不恰当锚点带来的负面影响，其精度高于多个展开的平均值。