  The ensemble Kalman inversion (EKI) is a particle based method which has been
introduced as the application of the ensemble Kalman filter to inverse
problems. In practice it has been widely used as derivative-free optimization
method in order to estimate unknown parameters from noisy measurement data. For
linear forward models the EKI can be viewed as gradient flow preconditioned by
a certain sample covariance matrix. Through the preconditioning the resulting
scheme remains in a finite dimensional subspace of the original
high-dimensional (or even infinite dimensional) parameter space and can be
viewed as optimizer restricted to this subspace. For general nonlinear forward
models the resulting EKI flow can only be viewed as gradient flow in
approximation. In this paper we discuss the effect of applying a sample
covariance as preconditioning matrix and quantify the gradient flow structure
of the EKI by controlling the approximation error through the spread in the
particle system. The ensemble collapse on the one side leads to an accurate
gradient approximation, but on the other side to degeneration in the
preconditioning sample covariance matrix. In order to ensure convergence as
optimization method we derive lower as well as upper bounds on the ensemble
collapse. Furthermore, we introduce covariance inflation without breaking the
subspace property intending to reduce the collapse rate of the ensemble such
that the convergence rate improves. In a numerical experiment we apply EKI to a
nonlinear elliptic boundary-value problem and illustrate the dependence of EKI
as derivative-free optimizer on the choice of the initial ensemble.
