The use of model order reduction techniques in combination with ensemble-based methods for estimating the state of systems described by nonlinear partial differential equations has been of great interest in recent years in the data assimilation community. Methods such as the multi-fidelity ensemble Kalman filter (MF-EnKF) and the multi-level ensemble Kalman filter (ML-EnKF) are recognized as state-of-the-art techniques. However, in many cases, the construction of low-fidelity models in an offline stage, before solving the data assimilation problem, prevents them from being both accurate and computationally efficient. In our work, we investigate the use of adaptive reduced basis techniques in which the approximation space is modified online based on the information that is extracted from a limited number of full order solutions and that is carried by the past models. This allows to simultaneously ensure good accuracy and low cost for the employed models and thus improve the performance of the multi-fidelity and multi-level methods.

翻译：近年来，在数据同化领域，将模型降阶技术与基于集成的方法相结合，用于估计由非线性偏微分方程描述的系统的状态，引起了极大的兴趣。多保真度集成卡尔曼滤波器（MF-EnKF）和多层级集成卡尔曼滤波器（ML-EnKF）等方法被认为是先进的技术。然而，在许多情况下，在求解数据同化问题之前的离线阶段构建低保真度模型，使得这些模型难以同时保证高精度和计算效率。在我们的工作中，我们研究了自适应降基技术的应用，其中近似空间是在线调整的，调整依据是从有限数量的全阶解中提取的信息以及由过往模型携带的信息。这能够同时确保所用模型具有良好的精度和较低的成本，从而提升多保真度和多层级方法的性能。