textabstractIn this article, we propose the use of partitioning and clustering methods as an alternative to Gaussian quadrature for stochastic collocation (SC). The key idea is to use cluster centers as the nodes for collocation. In this way, we can extend the use of collocation methods to uncertainty propagation with multivariate, correlated input. The approach is particularly useful in situations where the probability distribution of the input is unknown, and only a sample from the input distribution is available. We examine several clustering methods and assess their suitability for stochastic collocation numerically using the Genz test functions as benchmark. The proposed methods work well, most notably for the challenging case of nonlinearly correlated inputs in higher dimensions. Tests with input dimension up to 16 are included. Furthermore, the clustering-based collocation methods are compared to regular SC with tensor grids of Gaussian quadrature nodes. For 2-dimensional uncorrelated inputs, regular SC performs better, as should be expected, however the clustering-based methods also give only small relative errors. For correlated 2-dimensional inputs, clustering-based collocation outperforms a simple adapted version of regular SC, where the weights are adjusted to account for input correlation

中文翻译：

---

多变量相关输入不确定性传播的基于聚类的搭配

textabstract 在本文中，我们建议使用分区和聚类方法作为随机搭配 (SC) 高斯正交的替代方法。关键思想是使用集群中心作为节点进行搭配。通过这种方式，我们可以将搭配方法的使用扩展到具有多元相关输入的不确定性传播。该方法在输入的概率分布未知且只有输入分布中的样本可用的情况下特别有用。我们检查了几种聚类方法，并使用 Genz 测试函数作为基准，以数值方式评估它们对随机搭配的适用性。所提出的方法效果很好，最显着的是对于更高维度的非线性相关输入的具有挑战性的情况。包括输入维度高达 16 的测试。此外，将基于聚类的搭配方法与具有高斯正交节点张量网格的常规 SC 进行比较。对于二维不相关输入，常规 SC 表现更好，正如预期的那样，但是基于聚类的方法也只给出很小的相对误差。对于相关的二维输入，基于聚类的搭配优于常规 SC 的简单适应版本，其中调整权重以考虑输入相关性