Ensemble Kalman Inversion (EKI)
Initial Work
My work on Ensemble Kalman Inversion (EKI) originated with the paper [1], where concepts from the Ensemble Kalman Filter (EnKF), introduced by Evensen, were reformulated as an iterative solver for PDE-constrained inverse problems. Although similar ensemble ideas had previously been used for parameter identification in oil reservoir modelling, this work provided a unifying inverse problems perspective. As a result, EKI gained broader visibility within the inverse problems community and has since grown into an independent and active research direction.
EKI from an Iterative Regularisation Perspective
The initial work in [1] showed that a straightforward application of the EnKF as an iterative solver can suffer from stability issues similar to those encountered in classical iterative methods for ill-posed inverse problems: although the data misfit may decrease, the reconstruction error can grow. This observation motivated my subsequent work in [2], where ideas from Hanke’s regularising Levenberg–Marquardt method were adapted to modify EKI through the introduction of a regularisation parameter. From a heuristic perspective, this modification can be viewed as motivating EKI as a derivative-free approximation of the Levenberg–Marquardt method, in which ensemble-based covariances play the role of approximating local sensitivity information. The resulting scheme enables smoother updates and provides a principled stopping criterion. Although no theoretical analysis was developed at the time, the numerical results were very encouraging.
EKI from a Bayesian Perspective.
From a Bayesian perspective, EKI can be heuristically motivated using ideas from sequential Monte Carlo methods, where tempering is employed to construct a sequence of intermediate measures bridging the prior and the posterior. Within this viewpoint, EKI may be interpreted as replacing each intermediate posterior by a Gaussian approximation, linearising the forward map, and substituting explicit derivative information with ensemble-based covariance estimates. In the works [3], [4], we adopted this perspective to derive alternative, adaptive choices of the regularisation parameter in EKI, leading to methods that are robust and stable while still achieving fast convergence.