In 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 correlatio