  Ensemble Kalman inversion (EKI) is a derivative-free optimization method that
lies between the deterministic and the 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 $l_2$ penalty term, preventing overfitting in the standard
EKI. This paper proposes a strategy to implement $l_p, 0<p\leq 1,$
regularization for EKI to recover sparse structures in the solution. The
strategy transforms a $l_p$ problem into a $l_2$ 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 validate the proposed
approach's effectiveness and robustness through a suite of numerical
experiments, including compressive sensing and subsurface flow inverse
problems.
