A multi-fidelity ensemble Kalman filter with hyperreduced reduced-order models

We present an efficient data assimilation framework for nonlinear dynamical systems that uses multi-fidelity statistical estimates based on a full-order model and a projection-based hyperreduced order model (ROM) that is trained on-the-fly. This framework is particularly applicable to conservation laws in aerodynamics that yield expensive forward models. The formulation comprises the following technical components: (i) an ensemble Kalman filter to tractably handle high-dimensional, strongly nonlinear dynamical models; (ii) multi-fidelity forecast models, where ROM-based coarse fidelities are constructed on-the-fly; and (iii) hyperreduction for the ROM, based on proper orthogonal decomposition and the empirical quadrature procedure, constructed using the ensemble of full order model trajectories. We show that the multi-fidelity statistical estimates based on efficient, on-the-fly construction of the ROM enables rapid and reliable state estimation for practical nonlinear dynamical systems. We demonstrate the effectiveness of our framework to estimate the state of a separated flow around an airfoil by (i) showing that the ROM-based multi-fidelity method is more accurate than the single-fidelity method for comparable computational cost and (ii) showing that multi-fidelity statistics enable the use of a less accurate surrogate when compared against ROM-only filters, at negligible cost.