Higher-order moments of spline chaos expansion

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.

样条混沌展开，简称SCE，是输出随机变量依据输入随机变量中测度一致的正交归一化样条以及确定性系数的有限级数表示。本文报告了在计算输出随机变量（如果存在）的高阶矩时对SCE近似质量评估的新结果。提供了一种新的数学证明，以表明对于任意阶数的样条，随着最大单元尺寸减小，任意阶数的SCE矩收敛于精确矩。进行了补充的数值分析，其结果与理论发现一致。给出了一组简单但相关的示例，以将SCE的近似质量与著名的多项式混沌展开（PCE）进行比较。这些示例的结果表明，在本研究考虑的所有情况下，使用SCE计算的高阶矩都收敛。相比之下，大于二阶的PCE矩可能收敛，也可能不收敛，这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SCE和PCE生成的矩都收敛时，在输入随机变量存在非光滑函数或无界域的情况下，前者的收敛速度明显快于后者。