An Adaptive Hierarchical Ensemble Kalman Filter with Reduced Basis Models\ast

The use of reduced order modeling 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 multifidelity ensemble Kalman filter and the multilevel ensemble Kalman filter are recognized as state-of-the-art techniques. However, in many cases, the construction of low-fidelity models in an ovine 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 by combining information extracted from a limited number of full order solutions and information extracted from reduced models trained at previous time steps. This allows us to simultaneously ensure good accuracy and low cost for the employed models and thus improve the performance of the multifidelity and multilevel methods.