SIAM/ASA Journal on Uncertainty Quantification, Volume 8, Issue 1, Page 139-169, January 2020.
For the purpose of uncertainty propagation a new quadrature rule technique is proposed that has positive weights, has high degree, and is constructed using only samples that describe the probability distribution of the uncertain parameters. Moreover, nodes can be added to the quadrature rule, resulting in a sequence of nested rules. The rule is constructed by iterating over the samples of the distribution and exploiting the null space of the Vandermonde system that describes the nodes and weights, in order to select which samples will be used as nodes in the quadrature rule. The main novelty of the quadrature rule is that it can be constructed using any number of dimensions, using any basis, in any space, and using any distribution. It is demonstrated both theoretically and numerically that the rule always has positive weights and therefore has high convergence rates for sufficiently smooth functions. The convergence properties are demonstrated by approximating the integral of the Genz test functions. The applicability of the quadrature rule to complex uncertainty propagation cases is demonstrated by determining the statistics of the flow over an airfoil governed by the Euler equations, including the case of dependent uncertain input parameters. The new quadrature rule significantly outperforms classical sparse grid methods.

中文翻译：

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根据任意样本集生成具有正权重的嵌套正交规则

SIAM / ASA不确定性量化期刊，第8卷，第1期，第139-169页，2020年1月。
出于不确定性传播的目的，提出了一种新的正交规则技术，该技术具有正的权重，高度的程度，并且仅使用描述不确定性参数的概率分布的样本进行构造。此外，可以将节点添加到正交规则，从而产生一系列嵌套规则。通过遍历分布的样本并利用描述节点和权重的Vandermonde系统的空空间来构造规则，以便选择将哪些样本用作正交规则中的节点。正交规则的主要新颖之处在于，可以使用任意数量的维度，任何基础，任何空间以及任何分布来构造正交规则。从理论和数值上都证明了该规则始终具有正权重，因此对于足够平滑的函数而言具有较高的收敛速度。通过近似Genz测试函数的积分来证明收敛性。通过确定由Euler方程控制的翼型上的流动统计量（包括相关不确定输入参数的情况），证明了正交规则对复杂不确定性传播情况的适用性。新的正交规则大大优于经典的稀疏网格方法。通过确定由Euler方程控制的翼型上的流动统计量（包括相关不确定输入参数的情况），证明了正交规则对复杂不确定性传播情况的适用性。新的正交规则大大优于经典的稀疏网格方法。通过确定由Euler方程控制的翼型上的流动统计量（包括相关不确定输入参数的情况），证明了正交规则对复杂不确定性传播情况的适用性。新的正交规则大大优于经典的稀疏网格方法。