Polynomial chaos expansion has received considerable attention in uncertainty quantification since its great modeling capability for complex systems. However, considering the different variable distribution types and the ‘curse of dimensionality’ of the expansion coefficients, the polynomial chaos expansion has some limitations in the practical engineering application. In this paper, an optimal sparse polynomial chaos expansion is proposed to effectively realize uncertainty quantification with arbitrary probability distribution. This polynomial chaos expansion adopts the derivative $\lambda$-PDFs and the corresponding Gegenbauer polynomials to implement stochastic characterization for system model. Because of the excellent approximation ability of derivative $\lambda$-PDFs for arbitrary mono-peak distribution, the proposed polynomial chaos expansion has a wide adaptability to arbitrary distribution. Further, through the structure-selection technique based on error reduction ratio to screen the significant basis polynomials, the optimal sparse polynomial chaos expansion can be built with much fewer model evaluations than unknown coefficients. In view of that, the optimal sparse polynomial chaos expansion is applied to global sensitivity analysis, and the analytical solution of Sobol’ sensitivity indices is derived from the coefficients. Because of the advantages of simple structure, high precision and anti-noise ability, this optimal sparse polynomial chaos can improve the efficiency and accuracy of sensitivity analysis and has better adaptability for complex practical engineering problems. One numerical example and two engineering applications are studied to illustrate the practicability and effectiveness of the proposed method.

多项式混沌扩展因其对复杂系统的强大建模能力而在不确定性量化中受到了相当大的关注。然而，考虑到不同的变量分布类型和展开系数的“维数诅咒”，多项式混沌展开在实际工程应用中存在一定的局限性。为了有效地实现任意概率分布的不确定性量化，本文提出了一种最优稀疏多项式混沌展开。这个多项式混沌展开采用导数$\lambda$-PDF 和相应的 Gegenbauer 多项式来实现系统模型的随机表征。由于导数具有极好的逼近能力$\lambda$-PDFs对于任意单峰分布，提出的多项式混沌展开对任意分布具有广泛的适应性。此外，通过基于误差减少率的结构选择技术来筛选重要的基多项式，可以建立最优的稀疏多项式混沌扩展，而模型评估比未知系数。鉴于此，将最优稀疏多项式混沌展开应用于全局敏感性分析，并从系数中推导出Sobol's敏感性指标的解析解。由于结构简单、精度高、抗噪声能力强等优点，这种最优稀疏多项式混沌可以提高灵敏度分析的效率和准确性，对复杂的实际工程问题具有更好的适应性。通过一个数值算例和两个工程应用来说明该方法的实用性和有效性。