l1 regularization for Ensemble Kalman Inversion

Ensemble Kalman Inversion (EKI) is a sampling-based method for inverse problems, which is derivative-free and can utilize parallelization for efficiency. EKI uses a subspace property to regularize the inverse problems and can further implement Tikhonov regularization. Among other types of regularization methods, L1 regularization finds many applications, such as recovering sharp or sparse images, but its implementation in EKI has not yet been studied. This work proposes a strategy to impose L1 regularization in EKI. The modification of the EKI algorithm by the new method is minimal, and the method maintains a computational cost comparable to Tikhonov regularization. We provide a suite of numerical tests, including image reconstruction, to validate the proposed strategy's robustness and effectiveness.