Ensemble Kalman inversion with non-smooth regularization

TL;DR

This paper extends ensemble Kalman inversion to handle non-smooth regularization by developing a subgradient-based framework, establishing well-posedness, and demonstrating its effectiveness in inverse problems like computed tomography and sparse recovery.

Contribution

It introduces a subgradient-based formulation of EKI for non-smooth regularizers, providing theoretical analysis and practical algorithms for inverse problems.

Findings

Stable incorporation of non-smooth regularization in EKI.

Convergence analysis of the discrete-time scheme.

Successful numerical experiments in tomography and sparse recovery.

Abstract

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.