Adaptive regularisation for ensemble Kalman inversion

TL;DR

This paper introduces an adaptive regularisation method for ensemble Kalman inversion that automatically chooses parameters and determines early stopping, improving efficiency and robustness in inverse problems like electrical impedance tomography.

Contribution

It presents a novel adaptive regularisation strategy for EKI that does not rely on tuning parameters and controls divergence between measures, enhancing practical performance.

Findings

Efficient and robust estimates for both smooth and discontinuous fields.

Improved convergence and accuracy with the proposed regularisation.

Demonstrated computational efficiency in electrical impedance tomography.

Abstract

We propose a new regularisation strategy for the classical ensemble Kalman inversion (EKI) framework. The strategy consists of: (i) an adaptive choice for the regularisation parameter in the update formula in EKI, and (ii) criteria for the early stopping of the scheme. In contrast to existing approaches, our parameter choice does not rely on additional tuning parameters which often have severe effects on the efficiency of EKI. We motivate our approach using the interpretation of EKI as a Gaussian approximation in the Bayesian tempering setting for inverse problems. We show that our parameter choice controls the symmetrised Kulback-Leibler divergence between consecutive tempering measures. We further motivate our choice using a heuristic statistical discrepancy principle. We test our framework using electrical impedance tomography with the complete electrode model. Parameterisations of the unknown conductivity are employed which enable us to characterise both smooth or a discontinuous (piecewise-constant) fields. We show numerically that the proposed regularisation of EKI can produce efficient, robust and accurate estimates, even for the discontinuous case which tends to require larger ensembles and more iterations to converge. We compare the proposed technique with a standard method of choice and demonstrate that the proposed method is a viable choice to address computational efficiency of EKI in practical/operational settings.