Higher-order moments of spline chaos expansion混沌样条展开的高阶矩
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Rahman S, 2024. Higher-order moments of spline chaos expansion. Probabilistic Engineering Mechanics, 77: 103666.DOI: 10.1016/j.probengmech.2024.103666
混沌样条展开 (spline chaos expansion, SCE) 是一类有限级数表示方法，通过输入随机变量的测度一致正交样条和确定性系数来表示输出随机变量。本文介绍了混沌样条展开在计算输出随机变量高阶矩 (若存在) 近似值的新研究成果。提供了一种新的数学证明，表明无论样条阶数如何，混沌样条展开的任意阶矩随最大单元尺寸减小逐渐收敛至精确矩。此外，补充了数值分析，结果与理论一致。本文还给出了一组简单的相关算例，对比混沌样条展开与经典混沌多项式展开 (polynomial chaos expansion, PCE) 的近似值。算例结果表明，混沌样条展开计算的高阶矩在本研究所有算例中均能收敛。而混沌多项式展开的二阶以上矩是否收敛则取决于输出函数的正则性或输入随机变量的概率测度。此外，当混沌样条展开和混沌多项式展开生成的矩均收敛时，对于不光滑函数或输入随机变量无界域的情形，混沌样条展开的收敛速度明显快于混沌多项式展开。关键词: 不确定性量化, 基样条, 正交样条, 方差, 偏度, 峰度, 混沌多项式展开, 光滑模量, 误差分析Spline chaos expansion, referred to as SCE, is a finite series representation of an output random variable in terms of measure-consistent orthonormal splines in input random variables and deterministic coefficients. This paper reports new results from an assessment of SCE’s approximation quality in calculating higher-order moments, if they exist, of the output random variable. A novel mathematical proof is provided to demonstrate that the moment of SCE of an arbitrary order converges to the exact moment for any degree of splines as the largest element size decreases. Complementary numerical analyses have been conducted, producing results consistent with theoretical findings. A collection of simple yet relevant examples is presented to grade the approximation quality of SCE with that of the well-known polynomial chaos expansion (PCE). The results from these examples indicate that higher-order moments calculated using SCE converge for all cases considered in this study. In contrast, the moments of PCE of an order larger than two may or may not converge, depending on the regularity of the output function or the probability measure of input random variables. Moreover, when both SCE- and PCE-generated moments converge, the convergence rate of the former is markedly faster than the latter in the presence of nonsmooth functions or unbounded domains of input random variables.
Keywords: Uncertainty quantification; B-splines; Orthonormal splines; Variance; Skewness; Kurtosis; Polynomial chaos expansion; Modulus of smoothness; Error analysis.![]()
图 1: 给定结点序列下的一组二次基样条: (a) 非正交基; (b) Gauss 测度正交基; (c) 均匀测度正交基; (d) β 测度正交基Fig. 1. A set of B-splines associated with the knot sequence ξ = {-1,-1,-1,-0.5,0,0.5,1,1,1} and degree m = 2: (a) Non-orthonormal basis; (b) Orthonormal basis for uniform measure; (c) Orthonormal basis for truncated Gaussian measure; (d) Orthonormal basis for Beta measure
图 2: 双单变量函数与有界区间算例的一个光滑函数: (a) 不同阶次的混沌多项式展开近似; (b) 不同单元尺寸的线性混沌样条展开近似; (c) 不同单元尺寸的二次混沌样条展开近似; (d) 混沌样条展开与混沌多项式展开的方差相对误差; (e) 混沌样条展开与混沌多项式展开的偏度相对误差; (f) 混沌样条展开与混沌多项式展开的峰度相对误差Fig. 2. Smooth function of Example 1: y(X) = 1/(1+5X^2): (a) PCE approximations for m = 1, 2, 4, 8, 20; (b) Linear (m = 1) SCE approximations for h = 1, 1/2, 1/4, 1/8, 1/12; (c) Quadratic (m = 2) SCE approximations for h = 1, 1/2, 1/4, 1/8, 1/12; (d) Relative errors in variance from SCE (e_{2,m,h}) and PCE (e_{2,m}); (e) Relative errors in skewness from SCE (e_{3,m,h}) and PCE (e_{3,m}); (f) Relative errors in kurtosis from SCE (e_{4,m,h}) and PCE (e_{4,m})
图 3: 双单变量函数与有界区间算例的另一个光滑函数: (a) 不同阶次的混沌多项式展开近似; (b) 不同单元尺寸的线性混沌样条展开近似; (c) 不同单元尺寸的二次混沌样条展开近似; (d) 混沌样条展开与混沌多项式展开的方差相对误差; (e) 混沌样条展开与混沌多项式展开的偏度相对误差; (f) 混沌样条展开与混沌多项式展开的峰度相对误差Fig. 3. Smooth function of Example 1: y(X) = exp(-3|X|): (a) PCE approximations for m = 1, 2, 4, 8, 20; (b) Linear (m = 1) SCE approximations for h = 1, 1/2, 1/4, 1/8, 1/12, 1/16; (c) Quadratic (m = 2) SCE approximations for h = 1, 1/2, 1/4, 1/8, 1/12, 1/16; (d) Relative errors in variance from SCE (e_{2,m,h}) and PCE (e_{2,m}); (e) Relative errors in skewness from SCE (e_{3,m,h}) and PCE (e_{3,m}); (f) Relative errors in kurtosis from SCE (e_{4,m,h}) and PCE (e_{4,m})
图 4: 双单变量函数与无界区间算例的光滑函数: (a) 不同阶次的混沌多项式展开近似; (b) 不同单元尺寸的线性混沌样条展开近似; (c) 不同单元尺寸的二次混沌样条展开近似; (d) 混沌样条展开与混沌多项式展开的方差相对误差; (e) 混沌样条展开与混沌多项式展开的峰度相对误差Fig. 4. Smooth function of Example 2: y(X) = Φ(X): (a) PCE approximations for m = 1, 3, 5, 9, 21; (b) Linear (m = 1) SCE approximations for h = 8/3, 8/5, 8/7, 8/9, 8/13; (c) Quadratic (m = 2) SCE approximations for h = 8/3, 8/5, 8/7, 8/9, 8/13; (d) Relative errors in variance from SCE (e_{2,m,h}) and PCE (e_{2,m}); (e) Relative errors in kurtosis from SCE (e_{4,m,h}) and PCE (e_{4,m})
图 5: 双单变量函数与无界区间算例的非光滑函数: (a) 不同阶次的混沌多项式展开近似; (b) 不同单元尺寸的线性混沌样条展开近似; (c) 不同单元尺寸的二次混沌样条展开近似; (d) 混沌样条展开与混沌多项式展开的方差相对误差; (e) 混沌样条展开与混沌多项式展开的偏度相对误差; (f) 混沌样条展开与混沌多项式展开的峰度相对误差Fig. 5. Nonsmooth function of Example 2: y(X) = |X|: (a) PCE approximations for m = 2, 4, 8, 16, 24; (b) Linear (m = 1) SCE approximations for h = 8/3, 8/5, 8/7, 8/9, 8/13; (c) Quadratic (m = 2) SCE approximations for h = 8/3, 8/5, 8/7, 8/9, 8/13; (d) Relative errors in variance from SCE (e_{2,m,h}) and PCE (e_{2,m}); (e) Relative errors in skewness from SCE (e_{3,m,h}) and PCE (e_{3,m}); (f) Relative errors in kurtosis from SCE (e_{4,m,h}) and PCE (e_{4,m})
图 6: 非光滑双变量函数算例的函数图像: (a) 精确; (b) 20 阶混沌多项式展开; (c) 给定单元尺寸的的线性混沌样条展开; (d) 给定单元尺寸的的二次混沌样条展开Fig. 6. Graphs of functions in Example 3: (a) Exact; (b) 20th-order PCE; (c) Linear SCE with h = 1/10; (d) Quadratic SCE with h = 1/10
图 7: 非光滑双变量函数算例中非光滑函数各类混沌样条展开与混沌多项式展开近似的相对误差: (a) 方差; (b) 偏度; (c) 峰度Fig. 7. Relative errors from various SCE and PCE approximations of the nonmooth function in Example 3: (a) Variance (e_{2,m,h}, e_{2,m}); (b) Skewness (e_{3,m,h}, e_{3,m}); (c) Kurtosis (e_{4,m,h}, e_{4,m})
图 8: 三类方法估计的概率密度函数: (a) 混沌多项式展开; (b) 线性混沌样条展开; (c) 二次混沌样条展开Fig. 8. Probability density functions of y(X) estimated by three methods: (a) PCE; (b) Linear SCE; (c) Quadratic SCE
作者信息 | Authors
美国艾奥瓦大学 (University of Iowa) 工学院Email: sharif-rahman@uiowa.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)
