l_p regularization for ensemble Kalman inversion
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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 a compact set, 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≤ 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.
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