My work $l_p$ regularization for ensemble Kalman inversion is now published in SIAM Journal on Scientific Computing. The paper was accepted back in July 13, 2021.
Abstract: Ensemble Kalman inversion (EKI) is a derivative-free optimization method that lies between the deterministic and probabilistic approaches for inverse problems. EKI iterates the Kalman update of ensemble-based Kalman filters, whose ensemble converges to a minimizer of an objective function. EKI regularizes ill-posed problems by restricting the ensemble to the linear span of the initial ensemble, or by iterating regularization with early stopping. Another regularization approach for EKI, Tikhonov EKI, penalizes the objective function using the l2 penalty term, pre- venting overfitting in the standard EKI. This paper proposes a strategy to implement lp, 0 < p \leq 1, regularization for EKI to recover sparse structures in the solution. The strategy transforms an lp problem into an l2 problem, which is then solved by Tikhonov EKI. The transformation is explicit, and thus the proposed approach has a computational cost comparable to Tikhonov EKI. We vali- date the proposed approach’s effectiveness and robustness through a suite of numerical experiments, including compressive sensing and subsurface flow inverse problems.