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Bayesian uncertainty quantification (UQ) is of interest to industry andacademia as it provides a framework for quantifying and reducing theuncertainty in computational models by incorporating available data. Forsystems with very high computational costs, for instance, the computationalfluid dynamics (CFD) problem, the conventional, exact Bayesian approach such asMarkov chain Monte Carlo is intractable. To this end, the ensemble-basedBayesian methods have been used for CFD applications. However, theirapplicability for UQ has not been fully analyzed and understood thus far. Here,we evaluate the performance of three widely used iterative ensemble-based dataassimilation methods, namely ensemble Kalman filter, ensemble randomizedmaximum likelihood method, and ensemble Kalman filter with multiple dataassimilation for UQ problems. We present the derivations of the three ensemblemethods from an optimization viewpoint. Further, a scalar case is used todemonstrate the performance of the three different approaches with emphasis onthe effects of small ensemble sizes. Finally, we assess the three ensemblemethods for quantifying uncertainties in steady-state CFD problems involvingturbulent mean flows. Specifically, the Reynolds averaged Navier--Stokes (RANS)equation is considered the forward model, and the uncertainties in thepropagated velocity are quantified and reduced by incorporating observationdata. The results show that the ensemble methods cannot accurately capture thetrue posterior distribution, but they can provide a good estimation of theuncertainties even when very limited ensemble sizes are used. Based on theoverall performance and efficiency from the comparison, the ensemble randomizedmaximum likelihood method is identified as the best choice of approximateBayesian UQ approach~among the three ensemble methods evaluated here.

Evaluation of ensemble methods for quantifying uncertainties in steady-state CFD applications with small ensemble sizes