Quadratic point estimate method for probabilistic moments computation概率矩计算的二次点估计法
Ko MH, Papakonstantinou KG, 2024. Quadratic point estimate method for probabilistic moments computation. Probabilistic Engineering Mechanics, 79: 103705.DOI: 10.1016/j.probengmech.2024.103705
本文详细介绍了原创发展的二次点估计方法 (quadratic point estimate method, QPEM)，旨在采用 2n^2 + 1 个样本 (或 σ) 点高效精确地计算概率分布的前四阶输出矩，其中 n 为输入随机变量数。所提二次点估计方法特别为低维和中高维问题的现有采样和求积方法提供了有效精确且实用的替代方案。给出了详细的理论推导，证明所提方法在对称输入分布下可达到五阶以上精度。从简单多项式函数到随机场非线性有限元分析的各种数值算例验证了理论发现，并进一步展示了二次点估计方法从低维到高维问题的精度、效率和适用性。关键词: 不确定性传播, 点估计法, σ 点, 概率矩积分, 确定性抽样, 数值积分This paper presents in detail the originally developed Quadratic Point Estimate Method (QPEM), aimed at efficiently and accurately computing the first four output moments of probabilistic distributions, using 2n^2 + 1 sample (or sigma) points, with n, the number of input random variables. The proposed QPEM particularly offers an effective, superior, and practical alternative to existing sampling and quadrature methods for low- and moderately-high-dimensional problems. Detailed theoretical derivations are provided proving that the proposed method can achieve a fifth or higher-order accuracy for symmetric input distributions. Various numerical examples, from simple polynomial functions to nonlinear finite element analyses with random field representations, support the theoretical findings and further showcase the validity, efficiency, and applicability of the QPEM, from low- to high-dimensional problems.Keywords: Uncertainty propagation; Point estimate method; Sigma points; Probabilistic moment integrals; Deterministic sampling; Numerical integration.Fig. 1. Schematic representation of illustrative 2n + 1 PEM
Fig. 2. Schematic representation of the QPEM
Fig. 3. Stability factor of the QPEM
Fig. 4. Relative errors for the illustrative, discontinuous function example
Fig. 5. Moments estimation for polynomial functions with varying dimensions
Fig. 6. Schematic drawing of the roof structure
Fig. 7. Relative errors for the roof truss structure example
Fig. 8. Elastic bar with random field axial rigidity
Fig. 9. Relative errors for the elastic bar example
Fig. 10. Quarter-plate with a circular hole and random field elastic modulus
![]()
Fig. 11. Relative errors for the plate with a hole example
![]()
Fig. 12. Simply supported beam with material nonlinearity and random field elastic modulus
![]()
Fig. 13. Relative errors for the simply supported beam example
作者信息 | Authors
美国宾夕法尼亚州立大学 (Pennsylvania State University) 土木与环境工程系
Konstantinos G. Papakonstantinou, 通讯作者 (Corresp.)
美国宾夕法尼亚州立大学 (Pennsylvania State University) 土木与环境工程系Email: kpapakon@psu.edu
律梦泽 M.Z. Lyu | 编辑 (Ed)
P.D. Spanos | 审校 (Rev)
陈建兵 J.B. Chen | 审校 (Rev)
彭勇波 Y.B. Peng | 审校 (Rev)
