The use of model order reduction techniques in combination with data assimilation methods for estimating the state of systems has been of great interest in recent years. In this talk, we discuss some of our recent work in this area, focusing on three topics. In the first part, we focus on the ensemble Kalman method (EnKM) , an iterative Monte Carlo method for the solution of inverse problems. We show how model order reduction can be combined with the EnKM to greatly accelerate the EnKM solution of asynchronous data assimilation problems. In addition, we experimentally study the latter's performance with respect to different levels of noise and surrogate model error. Such numerical experiments, e.g., involving unknown distributed parameters in two or more spatial dimensions and testing several values o the inverse problem hyper-parameters. can be very expensive and are (here enabled only by the computational efficiency of the surrogate models. In the second part, we focus on methods such as the multi-fidelity ensemble Kalman filter (MFEnKF)  and the multi-level ensemble Kalman filter (MLEnKF)  for the solution of synchronous data assimilation problems. For these methods, the construction of low-fidelity models in the offline stage leads to a trade-off between accuracy and computational cost of the approximate models. In our work. we investigate the use of adaptive reduced-basis techniques in which the approximation space is modified (but not retrained) online based on the information extracted from the full-order solutions This has the potential to simultaneously ensure good accuracy and low cost for the employed models and thus improve the performance of the methods. Finally, we turn to Bayesian inversion, and consider some deterministic approaches for accounting for model error (e.g., resulting from the use of reduced order models) in inverse problems with additive Gaussian observation noise, where the parameter-to-observable map is the composition or a possibly nonlinear parameter-to-state map or 'model and a linear state-to-observable map or 'observatior operator.
Joint work of H. Bansal1, N. Cvetkovic1, M. Grepl2, H.C. Lie3, C. Pagliantini1, F.A.B. Silva1 and K. Veroy1
1Centre for Analysis, Scientific Computing and Applications, Eindhoven University of Technology (TUe), 5600 MB, Eindhoven, The Netherlands
2Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, 52056, Aachen, Germany
3Institut für Mathematik, Universität Potsdam, Potsdam OT Golm 14476, Germany
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Grant agreement No.818473).
 A. Chernov, H. Hoel, K. J. Law, F. Nobile, and R. Tempone. Multilevel ensemble kalman filtering for spatio-temporal processes. Numerische Mathematik, 147(1), 2021.
 M. A. Iglesias, K. J. H. Law, and A. M. Stuart. Ensemble kalman methods for inverse problems. Inverse Problems, 29, Mar 2013.
 A. A. Popov, C. Mou, A. Sandu, and T. Iliescu. A multifidelity ensemble kalman filter with reduced order control variates. SIAM Journal on Scientific Computing, 43(2), 2021.