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.

翻译：EKI 是一种无衍生的优化方法,它存在于确定性和概率方法之间,是解决反问题的确定性和概率方法之间。 EKI 将基于成堆的卡尔曼过滤器的全方位更新到卡尔曼更新,这些过滤器的共合性会与尽可能减少客观功能的最小化相融合。 EKI 将共和性限制在最初的组合的线性范围内,或通过及早中止转介正规化来规范不完善的问题。 EKI, Tikhonov EKI 的另一种正规化方法,利用$_2的罚款条款来惩罚目标功能,防止标准EKI的过度适用。本文提出了实施$_p, 0<pleq 1,使EKI的规范化成为一个最小化的功能,以恢复解决方案中的稀薄结构。 该战略将一个$_p$问题转化为$2美元的问题,然后由Tikhonov EKI解决。 改革是明确的,因此,拟议的方法的计算成本可与Tikhonov EKI 相仿,包括复合EKI。我们验证了拟议的数字和地面实验中的拟议方法的有效性和精确性。