Simple and efficient uncertainty propagation within numerical simulations involving low-dimensional random spaces

Abstract

This paper presents an uncertainty propagation method that synergistically combines discrete random variable transformations with support discretisation. Particularly suited for stochastic simulations of differential equations with parametric uncertainties in coefficients, initial, or boundary conditions, the method offers several key advantages: (1) it avoids random sampling; (2) it efficiently maps input to output probability distributions in a straightforward manner; (3) it remains nonintrusive, requiring no modification to the underlying deterministic model or its solution method; (4) it exhibits ease of parallelisation; and, (5) it uniquely allows for the reuse of information from prior analyses, to enhance convergence as well as to study variations in input distributions. While the individual concepts of support discretisation and discrete random variable transformations are established, their integrated application into this computationally efficient framework is underexplored. Hence, the novelty lies in presenting a detailed description of its simple methodology, demonstrating its application through four illustrative cases, and benchmarking its performance against prevalent sampling-based techniques. The results show that the approach achieves an accuracy comparable to that of other methods that use random sampling with, at least, an order of magnitude fewer realisations.