We propose a new regularisation strategy within the classical ensemble Kalman inversion (EKI) framework for estimating parameters in PDEs. The regularisation strategy consists of: (i) an adaptive choice for the regularisation /inflation parameter in the update formula in EKI, and (ii) criteria for the early stopping of the scheme. Our main contribution is the selection of the regularisation parameter which, in contrast to existing approaches, it does not rely on further parameters which often have severe effects on the efficiency of EKI. The proposed approach is aimed at providing EKI with computational efficiency and robustness for addressing problemswhere the unknown is a heterogeneous physical property with possibly sharp discontinuities arising from the presence of anomalies/defects. In these settings, for EKI to produce accurate estimates of the truth, the unknown needs to be suitably characterised via a parameterisation that often increases the complexity of the identification problem . We show numerically that the proposed approach can produce efficient, robust and accurate estimates under those challenging conditions which tend to require larger ensembles and more iterations to converge. We test our framework using various parameterisations including one that combines a truncated Whittle-Matern (WM) level-set function with other WM fields to characterise spatial variability of the physical property on each region. We use these parameterisations to solve PDE-constrained identification problems arising in (i) medical imaging where the aim is to detect the existence and properties of diseased tissue and (ii) non-destructive testing (NDT) of materials from data collected during manufacturing processes. We provide comparisons against a standard method  and demonstrate that the proposed method is  viable choice to address computational efficiency of EKI in practical/operational settings.
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.
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The proposed approach is aimed at providing EKI with computational efficiency and robustness for addressing problemswhere the unknown is a heterogeneous physical property with possibly sharp discontinuities arising from the presence of anomalies/defects. In these settings, for EKI to produce accurate estimates of the truth, the unknown needs to be suitably characterised via a parameterisation that often increases the complexity of the identification problem
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
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test our framework using various parameterisations including one that combines a truncated Whittle-Matern (WM) level-set function with other WM fields to characterise spatial variability of the physical property on each region. We use these parameterisations to solve PDE-constrained identification problems arising in (i) medical imaging where the aim is to detect the existence and properties of diseased tissue and (ii) non-destructive testing (NDT) of materials from data collected during manufacturing processes. We provide comparisons against
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