  The Ensemble Kalman filter (EnKF) was introduced by Evensen in 1994 [10] as a
novel method for data assimilation: state estimation for noisily observed
time-dependent problems. Since that time it has had enormous impact in many
application domains because of its robustness and ease of implementation, and
numerical evidence of its accuracy. In this paper we propose the application of
an iterative ensemble Kalman method for the solution of a wide class of inverse
problems. In this context we show that the estimate of the unknown function
that we obtain with the ensemble Kalman method lies in a subspace A spanned by
the initial ensemble. Hence the resulting error may be bounded above by the
error found from the best approximation in this subspace. We provide numerical
experiments which compare the error incurred by the ensemble Kalman method for
inverse problems with the error of the best approximation in A, and with
variants on traditional least-squares approaches, restricted to the subspace A.
In so doing we demonstrate that the ensemble Kalman method for inverse problems
provides a derivative-free optimization method with comparable accuracy to that
achieved by traditional least-squares approaches. Furthermore, we also
demonstrate that the accuracy is of the same order of magnitude as that
achieved by the best approximation. Three examples are used to demonstrate
these assertions: inversion of a compact linear operator; inversion of
piezometric head to determine hydraulic conductivity in a Darcy model of
groundwater flow; and inversion of Eulerian velocity measurements at positive
times to determine the initial condition in an incompressible fluid.
