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We propose the application of iterative regularization for the development ofensemble methods for solving Bayesian inverse problems. In concrete, weconstruct (i) a variational iterative regularizing ensemble Levenberg-Marquardtmethod (IR-enLM) and (ii) a derivative-free iterative ensemble Kalman smoother(IR-ES). The aim of these methods is to provide a robust ensemble approximationof the Bayesian posterior. The proposed methods are based on fundamental ideasfrom iterative regularization methods that have been widely used for thesolution of deterministic inverse problems [21]. In this work we are interestedin the application of the proposed ensemble methods for the solution ofBayesian inverse problems that arise in reservoir modeling applications. Theproposed ensemble methods use key aspects of the regularizingLevenberg-Marquardt scheme developed by Hanke [16] and that we recently appliedfor 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 randomizedmaximum likelihood (RML) and the ensemble smoother (ES) respectively. For thegeneral nonlinear case, we develop a numerical framework to assess theperformance of the proposed ensemble methods at capturing the posterior. Thisframework consists of using a state-of-the art MCMC method for resolving theBayesian posterior from synthetic experiments. The resolved posterior via MCMCthen provides a gold standard against to which compare the proposed IR-enLM andIR-ES. We show that for the careful selection of regularization parameters,robust approximations of the posterior can be accomplished in terms of mean andvariance. Our numerical experiments showcase the advantage of using iterativeregularization for obtaining more robust and stable approximation of theposterior than standard unregularized methods.

Iterative regularization for ensemble data assimilation in reservoir models