In this paper we discuss a deterministic form of ensemble Kalman inversion as a regularization method for linear inverse problems. By interpreting ensemble Kalman inversion as a low-rank approximation of Tikhonov regularization, we are able to introduce a new sampling scheme based on the Nystr\"om method that improves practical performance. Furthermore, we formulate an adaptive version of ensemble Kalman inversion where the sample size is coupled with the regularization parameter. We prove that the proposed scheme yields an order optimal regularization method under standard assumptions if the discrepancy principle is used as a stopping criterion. The paper concludes with a numerical comparison of the discussed methods for an inverse problem of the Radon transform.

翻译：本文探讨了作为线性逆问题正则化方法的集成卡尔曼逆变换的确定性形式。通过将集成卡尔曼逆变换解释为吉洪诺夫正则化的低秩近似，我们引入了一种基于Nyström方法的新采样方案，有效提升了实际性能。进一步地，我们提出了样本量与正则化参数耦合的自适应集成卡尔曼逆变换版本，并证明在采用偏差原理作为停止准则的标准假设下，该方案能实现阶最优的正则化方法。最后，通过Radon变换逆问题的数值实验对所述方法进行了比较。