Major Activities

The field of data assimilation seeks to find the best estimate for the unknown states or parameters in a dynamical system by combining appropriate mathematical models with observations and balancing the uncertainties. The linear Kalman filter, with extensions and generalizations such as the Extended Kalman filter, Unscented Kalman filter and Ensemble Kalman filter, has emerged as one of the the most popular data assimilation tool over the past few decades. In particular, the Ensemble Kalman filter (EnKF) has been developed as a recursive filter suitable for state-parameter estimation problems involving high dimensional PDEs. This project presents new contributions to the derivation and understanding of the Ensemble Kalman filter (EnKF) for inverse problems. Further, we provide a new proof on the convergence of recently developed Ensemble Kalman Inversion Algorithm (EnKI) from a functional analysis perspective and show that the new perspective allows one to develop strategies that helps to improve the convergence. Note that the EnKI algorithm is a derivative-free method and could be even employed for training a deep neural network especially when dealing with non-differentiable loss functions.

Significant Results

The proposed convergence improvement strategy is demonstrated for an Initial Condition Inversion in an Advection-Diffusion Problem. The method is compared with the commonly used additive inflation strategy for convergence improvement and avoiding filter divergence. The results are shown in Figure 1 below. It is clear from Figure 1 that the proposed strategy yields better solution compared to an existing method.

Figure 1: a) Solution obtained by existing additive inflation strategy (Schillings et al.); b) True solution; c) Solution obtained by the proposed convergence improvement strategy.

Further, the results for 1D deconvolution problem comparing different convergence improvement strategies is shown in Figure 2. EnKI-MC and EnKI-HC refers to multiplicative covariance correction and hybrid covariance correction respectively developed as part of the method.

Figure 2: Performance of different variants of EnKI (N=20 ensembles, Iterations=1000) for 1D Deconvolution problem. Left to right: Reconstructed solution by different methods; Convergence of relative error w.r.t truth for different methods.