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

Bayesian uncertainty quantification (UQ) is of interest to industry and academia as it provides a framework for quantifying and reducing the uncertainty in computational models by incorporating available data. For systems with very high computational costs, for instance, the computational fluid dynamics (CFD) problem, the conventional, exact Bayesian approach such as Markov chain Monte Carlo is intractable. To this end, the ensemble-based Bayesian methods have been used for CFD applications. However, their applicability for UQ has not been fully analyzed and understood thus far. Here, we evaluate the performance of three widely used iterative ensemble-based data assimilation methods, namely ensemble Kalman filter, ensemble randomized maximum likelihood method, and ensemble Kalman filter with multiple data assimilation for UQ problems. We present the derivations of the three ensemble methods from an optimization viewpoint. Further, a scalar case is used to demonstrate the performance of the three different approaches with emphasis on the effects of small ensemble sizes. Finally, we assess the three ensemble methods for quantifying uncertainties in steady-state CFD problems involving turbulent mean flows. Specifically, the Reynolds averaged Navier--Stokes (RANS) equation is considered the forward model, and the uncertainties in the propagated velocity are quantified and reduced by incorporating observation data. The results show that the ensemble methods cannot accurately capture the true posterior distribution, but they can provide a good estimation of the uncertainties even when very limited ensemble sizes are used. Based on the overall performance and efficiency from the comparison, the ensemble randomized maximum likelihood method is identified as the best choice of approximate Bayesian UQ approach~among the three ensemble methods evaluated here.