Uncertainty quantification (UQ) in mathematical models is essential for accurately predicting system behavior under variability. This study provides guidance on method selection for reliable UQ across varied functional behaviors in engineering applications. Specifically, we compare several interpolation and approximation methods within a stochastic collocation (SC) framework, namely: generalized polynomial chaos (gPC), B-splines, shape-preserving (SP) splines, and central weighted essentially nonoscillatory (CWENO) interpolation, to reconstruct probability density functions (PDFs) and estimate statistical moments. These methods are assessed for both smooth and discontinuous functions, as well as for the solution of the 1-D Euler and shallow water equations. While gPC and interpolation B-splines perform well with smooth data, they produce oscillations near discontinuities. Approximation B-splines and SP splines, while avoiding oscillations, converge more slowly. In contrast, CWENO interpolation demonstrates high robustness, effectively capturing sharp gradients without oscillations, making it suitable for complex, discontinuous data. Overall, CWENO interpolation emerges as a versatile and effective approach for SC, particularly in handling discontinuities in UQ.

翻译：数学模型中的不确定性量化对于准确预测系统在变异性下的行为至关重要。本研究为工程应用中不同函数行为的可靠不确定性量化提供了方法选择指导。具体而言，我们在随机配置框架内比较了多种插值与逼近方法，包括：广义多项式混沌、B样条、保形样条以及中心加权本质无振荡插值，以重构概率密度函数并估计统计矩。这些方法针对光滑函数与间断函数，以及一维欧拉方程和浅水方程的求解进行了评估。广义多项式混沌和插值B样条在光滑数据上表现良好，但在间断点附近会产生振荡。逼近B样条和保形样条虽能避免振荡，但收敛速度较慢。相比之下，CWENO插值表现出高度的鲁棒性，能有效捕捉急剧梯度而不产生振荡，适用于复杂的间断数据。总体而言，CWENO插值成为随机配置中一种通用且有效的方法，尤其适用于处理不确定性量化中的间断问题。