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In this article, we propose the use of partitioning and clustering methods asan alternative to Gaussian quadrature for stochastic collocation. The key ideais to use cluster centers as the nodes for collocation. In this way, we canextend the use of collocation methods to uncertainty propagation withmultivariate, dependent input, in which the output approximation is piecewiseconstant on the clusters. The approach is particularly useful in situationswhere the probability distribution of the input is unknown, and only a samplefrom the input distribution is available. We examine several clustering methodsand assess the convergence of collocation based on these methods boththeoretically and numerically. We demonstrate good performance of the proposedmethods, most notably for the challenging case of nonlinearly dependent inputsin higher dimensions. Numerical tests with input dimension up to 16 areincluded, using as benchmarks the Genz test functions and a test case fromcomputational fluid dynamics (lid-driven cavity flow).

Clustering-based collocation for uncertainty propagation with multivariate dependent inputs