A Dynamic Mode Decomposition approach for Data Assimilation in the presence of parameter uncertainties
Creators: name:  A. VERVYNCK 
name:   V. MONS 
name:   F. CHAMPAGNAT 
Publisher: name: Proceedings of  the 9th European Conference for Aerospace Sciences. Lille, France, 27 June - 1 July, 2022
Year: 2022
Subjects: FLIGHT PHYSICS
COMPUTATIONAL FLUID DYNAMICS

description: Data assimilation allows combining experimental and numerical data to improve state prediction. However, it can be very costly if the state is of high order, and this applies in particular to ensemble methods such as the Ensemble Kalman filter (EnKF) [1]. Indeed the computational cost increases with the number of ensemble members which could mean multiple Computational Fluid Dynamics (CFD) calculations have to be done at each forecast step of the filter in the context of fluid flows. It can be thus very interesting to develop methodologies that demand low computational cost. Our approach combines EnKF and Dynamic Mode Decomposition (DMD) for approximate low rank linear state propagation. Each DMD is calculated offline from snapshots issued from a particular set of conditions, and a reduced order state can be built using the modes. Using DMD, one can reformulate a Kalman Filter in reduced order state with a diagonal forecast step. Analysis step can also be performed in reduced order state, thus the overall assimilation can be performed in a computationally efficient way. However, the drawback is that the state remains in the subspace that is spanned by the DMD modes and that a small error on a model parameter strongly deteriorates the results of the filter. This leads to the idea in this work which is to create a catalogue of different DMD and use all the modes generated to create a combined reduced order space. The state is then projected on all subspaces simultaneously and one assimilation is performed on all the modes. This is different from what has been done in [2] where each DMD is used to create a different subspace: multiple assimilations then work independently on the subspaces. Whereas their approach seeks to track a time-varying state evolution, our goal is to match a time-persistent state evolution with parameter uncertainties. The filter developed here is then initialised using a method developed in [1] in which the initial ensemble is generated randomly in the subspace generated by the modes. Several ensemble members are generated using each model in the catalogue in order to cover the subspace span by all approximate models. This approach retains the advantages of combining the EnKF and the DMD as it is still computationally efficient. In order to test this methodology, a 2D laminar flow around a circular cylinder and its wake is studied. The reference flow is generated at a given Reynolds number using a Direct Numerical Simulation and the measurements are extracted from this simulation. This Reynolds number is then assumed to be unknown and a catalogue of DMD is built using several CFD simulations at different Reynolds number surrounding that of the reference flow. We then show that if only one DMD model is used with a 2% error on the Reynolds number, the filter fails to correctly predict the state (despite adding inflation). This is unlike our approach for which we show that combining multiple DMD to form a new subspace, the results are similar to when the EnKF is done with a DMD based on the reference flow.