SIAM/ASA Journal on Uncertainty Quantification, Volume 8, Issue 1, Page 27-57, January 2020.
A spline chaos expansion, referred to as SCE, is introduced for uncertainty quantification analysis. The expansion provides a means for representing an output random variable of interest with respect to multivariate orthonormal basis splines (B-splines) in input random variables. The multivariate B-splines are built from a whitening transformation to generate univariate orthonormal B-splines in each coordinate direction, followed by a tensor-product structure to produce the multivariate version. SCE, as it stems from compactly supported B-splines, tackles locally prominent responses more effectively than the polynomial chaos expansion (PCE). The approximation quality of the expansion is demonstrated in terms of the modulus of smoothness of the output function, leading to the mean-square convergence of SCE to the correct limit. Analytical formulae are proposed to calculate the mean and variance of an SCE approximation for a general output variable in terms of the requisite expansion coefficients. Numerical results indicate that a low-order SCE approximation with an adequate mesh is markedly more accurate than a high-order PCE approximation in estimating the output variances and probability distributions of oscillatory, nonsmooth, and nearly discontinuous functions.

中文翻译：

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样条混沌扩展

SIAM / ASA不确定性量化杂志，第8卷，第1期，第27-57页，2020年1月。
引入样条混沌扩展，称为SCE，用于不确定性量化分析。展开提供了一种手段，用于表示与输入随机变量中的多元正交基样条（B样条）有关的感兴趣的输出随机变量。多元B样条是通过变白生成的，以在每个坐标方向上生成单变量正交B样条，然后是张量积结构以生成多元版本。SCE源于紧密支持的B样条曲线，比多项式混沌扩展（PCE）更有效地解决了局部突出的响应。扩展的近似质量以输出函数的平滑模数表示，从而导致SCE的均方收敛到正确的极限。根据所需的膨胀系数，提出了解析公式来计算一般输出变量的SCE近似值的均值和方差。数值结果表明，在估计振荡函数，非平滑函数和几乎不连续函数的输出方差和概率分布时，具有足够网格的低阶SCE近似比高阶PCE近似更为准确。