在稳态CFD应用中用小集合尺寸量化不确定性的集合方法的评价
Evaluation of ensemble methods for quantifying uncertainties in steady-state CFD applications with small ensemble sizes
摘要：
贝叶斯不确定性量化（UQ）是工业界和学术界感兴趣的，因为它提供了一个框架，量化和减少计算模型中的不确定性，结合可用的数据。对于具有非常高的计算成本的系统，例如，计算流体动力学（CFD）问题，常规的精确贝叶斯方法（诸如马尔可夫链蒙特卡罗）是难以处理的。为此，基于集合的贝叶斯方法已被用于CFD应用。然而，它们对UQ的适用性迄今尚未得到充分分析和理解。在这里，我们评估了三个广泛使用的迭代集成数据同化方法，即集合卡尔曼滤波，集合随机极大似然方法，集合卡尔曼滤波与多个数据同化的UQ问题的性能。我们提出了三个集成方法从优化的角度推导。此外，标量的情况下被用来证明三种不同的方法的性能，强调小合奏大小的影响。最后，我们评估了三个合奏方法量化的不确定性，在稳态CFD问题涉及湍流平均流。具体地说，雷诺平均Navier-Stokes（RANS）方程被认为是向前的模型，和传播速度的不确定性进行量化，并通过结合观测数据减少。结果表明，集成方法不能准确地捕捉真实的后验分布，但他们可以提供一个很好的估计的不确定性，即使在非常有限的集成规模使用。基于整体性能和效率的比较，集成随机极大似然方法被确定为近似贝叶斯UQ方法的最佳选择~在这里评估的三个集成方法。
英文摘要：
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.
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