Adaptive Ensemble Kalman Inversion with Statistical Linearization

The ensemble Kalman inversion (EKI), inspired by the well-known ensem-ble Kalman filter, is a derivative-free and parallelizable method for solving inverse problems. The method is appealing for applications in a variety of fields due to its low computational cost and simple implementation. In this paper, we propose an adaptive ensemble Kalman inversion with statistical linearization (AEKI-SL) method for solving inverse problems from a hierarchical Bayesian perspective. Specifically, by adaptively updating the unknown with an EKI and updating the hyper-parameter in the prior model, the method can improve the accuracy of the solutions to the inverse problem. To avoid semi-convergence, we employ Morozov's discrepancy principle as a stopping criterion. Furthermore, we extend the method to simultaneous estimation of noise levels in order to reduce the randomness of artificially ensemble noise levels. The convergence of the hyper-parameter in prior model is investigated theoretically. Numerical experiments show that our proposed methods outperform the traditional EKI and EKI with statistical linearization (EKI-SL) methods.