Ensemble Kalman inversion with non-smooth regularization
This work addresses a gap in EKI methods for inverse problems with non-smooth regularization, offering a novel framework that is incremental but improves applicability in domains like imaging and signal processing.
The paper tackled the challenge of incorporating non-smooth regularization into ensemble Kalman inversion (EKI) for variational inverse problems, introducing a subgradient-based formulation (SEKI) that demonstrated stable and principled performance in numerical experiments like computed tomography with total variation regularization and sparse recovery with ℓ1 penalties.
This paper investigates ensemble Kalman inversion (EKI) for variational inverse problems with convex, potentially non-smooth regularization. While deterministic EKI and its Tikhonov-regularized variants have primarily been analyzed for smooth objectives, a corresponding framework accommodating subgradient dynamics has not yet been established. To address this gap, we introduce a subgradient-based formulation of EKI (SEKI) that incorporates non-smooth regularizers through a covariance-preconditioned differential inclusion for the ensemble mean. In the linear forward-model setting, well-posedness of the resulting continuous-time particle system is established under minimal assumptions on the regularization functional using maximal monotone operator theory and Yosida approximations. Motivated by the continuous-time dynamics, we propose an explicit discrete-time scheme that preserves the derivative-free structure of EKI and analyze its convergence as an optimization method in the strongly convex case. Numerical experiments in computed tomography with total variation regularization and sparse recovery with $\ell_1$ penalties illustrate that non-smooth regularization can be incorporated into ensemble Kalman inversion in a stable and principled manner.