# Clustering-based collocation for uncertainty propagation with multivariate dependent inputs

## TL;DR

This paper introduces a clustering-based collocation method for uncertainty propagation with multivariate dependent inputs, offering an alternative to Gaussian quadrature especially when the input distribution is unknown.

## Contribution

It proposes using cluster centers as collocation nodes, extending stochastic collocation to dependent inputs, and evaluates the method's convergence both theoretically and numerically.

## Key findings

- Effective for high-dimensional, nonlinear dependent inputs
- Performs well with limited input distribution information
- Demonstrates good convergence and accuracy in numerical tests

## Abstract

In this article, we propose the use of partitioning and clustering methods as an alternative to Gaussian quadrature for stochastic collocation. 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, dependent input, in which the output approximation is piecewise constant on the clusters. 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 the convergence of collocation based on these methods both theoretically and numerically. We demonstrate good performance of the proposed methods, most notably for the challenging case of nonlinearly dependent inputs in higher dimensions. Numerical tests with input dimension up to 16 are included, using as benchmarks the Genz test functions and a test case from computational fluid dynamics (lid-driven cavity flow).