An optimal version of spline dimensional decomposition (SDD) is unveiled for general high-dimensional uncertainty quantification analysis of complex systems subject to independent but otherwise arbitrary probability measures of input random variables. The resulting method involves optimally derived knot vectors of basis splines (B-splines) in some or all coordinate directions, whitening transformation producing measure-consistent orthonormalized B-splines equipped with optimal knots, and Fourier-spline expansion of a general high-dimensional output function of interest. In contrast to standard SDD, there is no need to select the knot vectors uniformly or intuitively. The generation of optimal knot vectors can be viewed as an inexpensive preprocessing step toward creating the optimal SDD. Analytical formulas are proposed to calculate the second-moment properties by the optimal SDD method for a general output random variable in terms of the expansion coefficients involved. It has been shown that the computational complexity of the optimal SDD method is polynomial, as opposed to exponential, thus mitigating the curse of dimensionality by a discernible magnitude. Numerical results affirm that the optimal SDD method developed is more precise than polynomial chaos expansion, sparse-grid quadrature, and the standard SDD method in calculating not only the second-moment statistics, but also the cumulative distribution function of an output random variable. More importantly, the optimal SDD outperforms standard SDD by sustaining nearly identical computational cost.

中文翻译：

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最优样条维数分解的不确定性量化

样条维数分解 (SDD) 的最佳版本被公开，用于复杂系统的一般高维不确定性量化分析，这些系统受输入随机变量的独立但任意的概率度量影响。由此产生的方法涉及在某些或所有坐标方向上优化导出的基础样条（B 样条）的节点向量，白化变换产生配备最优节点的测量一致的正交化 B 样条，以及一般高维输出的傅立叶样条扩展兴趣函数。与标准 SDD 相比，无需统一或直观地选择节点向量。最佳结向量的生成可以被视为创建最佳 SDD 的廉价预处理步骤。根据所涉及的展开系数，针对一般输出随机变量，提出了通过最优SDD方法计算二阶矩性质的解析公式。已经表明，最优 SDD 方法的计算复杂度是多项式的，而不是指数的，从而以可辨别的幅度减轻维度灾难。数值结果证实，所开发的最优 SDD 方法比多项式混沌展开、稀疏网格正交和标准 SDD 方法更精确，不仅可以计算二阶矩统计量，而且可以计算输出随机变量的累积分布函数。更重要的是，最优的 SDD 通过维持几乎相同的计算成本来优于标准 SDD。