  We introduce a derivative-free computational framework for approximating
solutions to nonlinear PDE-constrained inverse problems. The aim is to merge
ideas from iterative regularization with ensemble Kalman methods from Bayesian
inference to develop a derivative-free stable method easy to implement in
applications where the PDE (forward) model is only accessible as a black box.
The method can be derived as an approximation of the regularizing
Levenberg-Marquardt (LM) scheme [14] in which the derivative of the forward
operator and its adjoint are replaced with empirical covariances from an
ensemble of elements from the admissible space of solutions. The resulting
ensemble method consists of an update formula that is applied to each ensemble
member and that has a regularization parameter selected in a similar fashion to
the one in the LM scheme. Moreover, an early termination of the scheme is
proposed according to a discrepancy principle-type of criterion. The proposed
method can be also viewed as a regularizing version of standard Kalman
approaches which are often unstable unless ad-hoc fixes, such as covariance
localization, are implemented. We provide a numerical investigation of the
conditions under which the proposed method inherits the regularizing properties
of the LM scheme of [14]. More concretely, we study the effect of ensemble
size, number of measurements, selection of initial ensemble and tunable
parameters on the performance of the method. The numerical investigation is
carried out with synthetic experiments on two model inverse problems: (i)
identification of conductivity on a Darcy flow model and (ii) electrical
impedance tomography with the complete electrode model. We further demonstrate
the potential application of the method in solving shape identification
problems by means of a level-set approach for the parameterization of unknown
geometries.
