Localized ensemble Kalman inversion

Ensemble Kalman inversion (EKI) is an adaption of the ensemble Kalman filter (EnKF) for the numerical solution of inverse problems. Both EKI and EnKF suffer from the 'subspace property', i.e. the EKI and EnKF solutions are linear combinations of the initial ensembles. The subspace property implies that the ensemble size should be larger than the problem dimension to ensure EKI's convergence to the correct solution. This scaling of ensemble size is impractical and prevents the use of EKI in high-dimensional problems. 'Localization' has been used for many years in EnKF to break the subspace property in a way that a localized EnKF can solve high-dimensional problems with a modest ensemble size, independently of the number of unknowns. Here, we study localization of the EKI and demonstrate how a localized EKI (LEKI) can solve high-dimensional inverse problems with a modest ensemble size. Our analysis is mathematically rigorous and applies to the continuous time limit of the EKI. Specifically, we can prove an intended ensemble collapse and convergence guarantees with an ensemble size that is less than the number of unknowns, which sets this work apart from the current state-of-the-art. We illustrate our theory with numerical experiments where some of our mathematical assumptions may only be approximately valid.