Adaptive regularisation for ensemble Kalman inversion

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 Kullback-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.

Iterative ensemble Kalman methods: A unified perspective with some new variants

“…However, we have observed in a variety of numerical examples (not reported here) that in nonlinear problems it is often necessary to run EKI up to times larger than 1 to obtain adequate approximation of the posterior mean and covariance. Providing a suitable stopping criteria for EKI is a topic of current research [56,32] beyond the scope of our work.…”

“…• An important methodological question, still largely unresolved, is the development of adaptive and easily implementable line search schemes and stopping criteria for ensemble-based optimization schemes. An important work in this direction is [32]. • Another avenue for future methodological research is the development of iterative ensemble Kalman methods that are sparse-promoting, considering alternative regularizations beyond the least-squares objectives discussed in our paper [38,40,57].…”

Ensemble Kalman Inversion for General Likelihoods

“…The effective sample size (over true importance weights) can be viewed as a measure of the χ 2 -divergence between the tempered posterior at λ −1 and λ [4]. In [11] this idea was extended to instead use an effective sample size based on the symmetric KL-divergence.…”

Localization in Ensemble Kalman inversion

“…There are many recent works that carefully describe the mathematics of EKI [5,10], that suggest improvements [11,12,13,14], and that explain how EKI can be used in machine learning [15]. An extension of EKI, such that the EKI ensemble is distributed according to a Bayesian posterior distribution, is discussed in [16] and is called the ensemble Kalman sampler.…”