  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.
