  We propose the application of iterative regularization for the development of
ensemble methods for solving Bayesian inverse problems. In concrete, we
construct (i) a variational iterative regularizing ensemble Levenberg-Marquardt
method (IR-enLM) and (ii) a derivative-free iterative ensemble Kalman smoother
(IR-ES). The aim of these methods is to provide a robust ensemble approximation
of the Bayesian posterior. The proposed methods are based on fundamental ideas
from iterative regularization methods that have been widely used for the
solution of deterministic inverse problems [21]. In this work we are interested
in the application of the proposed ensemble methods for the solution of
Bayesian inverse problems that arise in reservoir modeling applications. The
proposed ensemble methods use key aspects of the regularizing
Levenberg-Marquardt scheme developed by Hanke [16] and that we recently applied
for history matching in [18].
  In the case where the forward operator is linear and the prior is Gaussian,
we show that the proposed IR-enLM and IR-ES coincide with standard randomized
maximum likelihood (RML) and the ensemble smoother (ES) respectively. For the
general nonlinear case, we develop a numerical framework to assess the
performance of the proposed ensemble methods at capturing the posterior. This
framework consists of using a state-of-the art MCMC method for resolving the
Bayesian posterior from synthetic experiments. The resolved posterior via MCMC
then provides a gold standard against to which compare the proposed IR-enLM and
IR-ES. We show that for the careful selection of regularization parameters,
robust approximations of the posterior can be accomplished in terms of mean and
variance. Our numerical experiments showcase the advantage of using iterative
regularization for obtaining more robust and stable approximation of the
posterior than standard unregularized methods.
