标题： 样条混沌展开 显示英文标题

标题： A Spline Chaos Expansion

摘要： 一种称为SCE（样条混沌展开）的方法被引入用于不确定性量化分析。该展开提供了一种手段，可以借助输入随机变量的多元正交B样条（B-spline）来表示感兴趣的输出随机变量。多元B样条通过白化变换构建，生成每个坐标方向上的单变量正交B样条，然后通过张量积结构生成多元版本。由于SCE源自紧支集B样条，它比多项式混沌展开（PCE）更有效地处理局部显著响应。通过输出函数的光滑模数展示展开的近似质量，从而实现SCE向正确极限的均方收敛。提出了分析公式，以扩展系数的形式计算一般输出变量的SCE近似值的均值和方差。数值结果显示，对于估计振荡、非光滑且接近不连续函数的输出方差和概率分布，一个具有适当网格的低阶SCE近似明显比高阶PCE近似更准确。 显示英文摘要

摘要： 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.