A sequential estimator based on the Ensemble Kalman Filter for Data Assimilation of fluid flows is presented in this research work. The main feature of this estimator is that the Kalman filter update, which relies on the determination of the Kalman gain, is performed exploiting the algorithmic features of the numerical solver employed as a model. More precisely, the multilevel resolution associated with the multigrid iterative approach for time advancement is used to generate several low-resolution numerical simulations. These results are used as ensemble members to determine the correction via Kalman filter, which is then projected on the high-resolution grid to correct a single simulation which corresponds to the numerical model. The assessment of the method is performed via the analysis of one-dimensional and two-dimensional test cases, using different dynamical equations. The results show an efficient trade-off in terms of accuracy and computational costs required. In addition, a physical regularization of the flow, which is not granted by classical KF approaches, is naturally obtained owing to the multigrid iterative calculations. The algorithm is also well suited for the analysis of unsteady phenomena and, in particular, for potential application to in-streaming Data Assimilation techniques.

在这项研究工作中，提出了一种基于用于流体流动数据同化的集成卡尔曼滤波器的顺序估计器。该估计器的主要特征是，依赖于卡尔曼增益的确定的卡尔曼滤波器更新是利用用作模型的数值求解器的算法特征来执行的。更准确地说，与时间推进的多重网格迭代方法相关的多级分辨率用于生成几个低分辨率数值模拟。这些结果被用作集合成员，通过卡尔曼滤波器确定校正，然后将其投影到高分辨率网格上以校正对应于数值模型的单个模拟。该方法的评估是通过对一维和二维测试用例的分析来进行的，使用不同的动力学方程。结果显示在所需的准确性和计算成本方面进行了有效的权衡。此外，由于多重网格迭代计算，自然获得了经典 KF 方法无法实现的物理正则化。该算法也非常适合分析不稳定现象，特别是对流式数据同化技术的潜在应用。