  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.
