Sampling error correction in ensemble Kalman inversion

Ensemble Kalman inversion is a parallelizable derivative-free method to solve inverse problems. The method uses an ensemble that follows the Kalman update formula iteratively to solve an optimization problem. The ensemble size is crucial to capture the correct statistical information in estimating the unknown variable of interest. Still, the ensemble is limited to a size smaller than the unknown variable's dimension for computational efficiency. This study proposes a strategy to correct the sampling error due to a small ensemble size, which improves the performance of the ensemble Kalman inversion. This study validates the efficiency and robustness of the proposed strategy through a suite of numerical tests, including compressive sensing, image deblurring, parameter estimation of a nonlinear dynamical system, and a PDE-constrained inverse problem.