On convergence rates of adaptive ensemble Kalman inversion for linear ill-posed problems

In this paper we show that ensemble Kalman inversion for linear inverse problems can equivalently be formulated as a stochastic low-rank approximation of Tikhonov regularization. This point of view allows us to introduce an alternative 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 under standard assumptions that the proposed scheme yields an order optimal regularization method if the discrepancy principle is used as a stopping criterion.