A Spline Chaos Expansion
A Spline Chaos Expansion
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DOI：
10.1137/19m1239702
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发表时间：
2019-11
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影响因子：
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通讯作者：
S. Rahman
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文献类型：
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作者：
S. Rahman
引入了一种用于不确定量化分析的Spline混沌展开方法。该展开提供了一种关于输入随机变量中的多变量正交基样条(B-样条)来表示感兴趣的输出随机变量的手段。通过白化变换构造多元B-样条，在每个坐标方向上产生单变量的正交化B-样条，然后用张量积结构产生多元形式。与多项式混沌展开(PCE)相比，基于紧支撑的B-Spline的SCE能更有效地处理局部显著的响应。用输出函数的光滑模来证明展开式的逼近质量，从而使SCE的均方收敛到正确的极限。根据所需的展开系数，提出了计算一般输出变量的SCE近似的均值和方差的解析公式。数值结果表明，在估计振荡、非光滑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近似明显比高阶PCE近似精确得多。
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.