稀疏偏最小二乘回归-多项式混沌展开代理模型方法

为解决传统多项式混沌展开方法在高维全局灵敏度和结构可靠度分析当中存在的维数灾难与多重共线性问题，该文提出一种稀疏偏最小二乘回归-多项式混沌展开代理模型方法。该方法首先采用偏最小二乘回归技术得到多项式混沌展开系数的初步估计，然后根据回归误差阈值允许下的最大稀疏度原则，采用带有惩罚的矩阵分解技术自适应地保留与结构响应相关性强的多项式，并采用偏最小二乘回归得到多项式混沌展开系数的更新估计。通过对展开系数进行简单后处理即可得到Sobol灵敏度指数。在此基础上保留重要输入变量并按新方法重新进行回归可实现对代理模型的精简，从而在不增加计算代价的情况下实现高精度结构可靠度分析。算例结果表明在保证精度的情况下，采用新方法进行全局灵敏度和结构可靠度分析比传统方法在计算效率方面有显著优势。

SPARSE PARTIAL LEAST SQUARES REGRESSION-POLYNOMIAL CHAOS EXPANSION METAMODELING METHOD

To circumvent the curse of dimensionality and multicollinearity problems of traditional polynomial chaos expansion approach when analyzing global sensitivity and structural reliability of high-dimensional models, this paper proposes a sparse partial least squares regression-polynomial chaos expansion metamodeling method. Firstly, an initial estimation of polynomial chaos expansion coefficients is obtained with the partial least squares regression. Secondly, according to the principle of maximum sparsity under the allowance of regression error threshold, polynomials which have strong correlation with the structural response are adaptively retained with the penalized matrix decomposition scheme. Next, an updated estimation of the polynomial chaos expansion coefficients is obtained with the partial least squares regression. Sobol sensitivity indices are obtained with a simple post-processing of the expansion coefficients. Finally, the metamodel is greatly simplified by regressing with important inputs, leading to accurate estimations of the failure probability without additional computational cost. The results show that with acceptable accuracies, the new method overperforms the traditional counterpart in terms of computational efficiency when solving high-dimensional global sensitivity and structural reliability analysis problems.