$l_p$ regularization for ensemble Kalman inversion

TL;DR

This paper introduces an $l_p$ regularization strategy for ensemble Kalman inversion (EKI) to promote sparsity in solutions, transforming $l_p$ problems into $l_2$ problems solvable by Tikhonov EKI, with demonstrated effectiveness in numerical experiments.

Contribution

The paper proposes a novel explicit transformation method to implement $l_p$ regularization in EKI, enabling sparse solution recovery with computational costs similar to Tikhonov EKI.

Findings

Effective recovery of sparse structures demonstrated in compressive sensing tasks.

Robustness validated through subsurface flow inverse problem experiments.

Transformation approach maintains computational efficiency.

Abstract

Ensemble Kalman inversion (EKI) is a derivative-free optimization method that lies between the deterministic and the probabilistic approaches for inverse problems. EKI iterates the Kalman update of ensemble-based Kalman filters, whose ensemble converges to a minimizer of an objective function. EKI regularizes ill-posed problems by restricting the ensemble to the linear span of the initial ensemble, or by iterating regularization with early stopping. Another regularization approach for EKI, Tikhonov EKI, penalizes the objective function using the $l_2$ penalty term, preventing overfitting in the standard EKI. This paper proposes a strategy to implement $l_p, 0<p\leq 1,$ regularization for EKI to recover sparse structures in the solution. The strategy transforms a $l_p$ problem into a $l_2$ problem, which is then solved by Tikhonov EKI. The transformation is explicit, and thus the proposed approach has a computational cost comparable to Tikhonov EKI. We validate the proposed approach's effectiveness and robustness through a suite of numerical experiments, including compressive sensing and subsurface flow inverse problems.