[Paper Review] Adaptive regularisation for ensemble Kalman inversion with applications to non-destructive testing and imaging

Marco Iglesias, Yuchen Yang|arXiv (Cornell University)|Jun 26, 2020

Electrical and Bioimpedance Tomography42 references4 citations

TL;DR

This paper introduces an adaptive regularisation strategy for ensemble Kalman inversion (EKI) that dynamically adjusts the regularisation/inflation parameter and determines optimal stopping times, eliminating reliance on tuning parameters. The method enhances computational efficiency and robustness in estimating heterogeneous, discontinuous physical properties—critical for non-destructive testing and medical imaging—by combining truncated Whittle-Matern level-set parameterisations with PDE-constrained inversion, outperforming standard EKI in accuracy and convergence under challenging conditions.

ABSTRACT

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 problems where 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.

Motivation & Objective

• Address the challenge of estimating heterogeneous, discontinuous physical properties in PDE-constrained inverse problems, particularly in non-destructive testing and medical imaging.
• Overcome the limitations of standard EKI, which often requires large ensembles and many iterations due to poor regularisation and lack of robust stopping criteria.
• Develop a regularisation strategy that adapts the inflation parameter dynamically without introducing additional tuning parameters.
• Improve computational efficiency and convergence robustness in parameter estimation for problems with sharp discontinuities and complex spatial variability.
• Enable practical deployment of EKI in operational settings by ensuring reliable and accurate reconstructions under realistic data and model complexity.

Proposed method

• Introduce an adaptive regularisation scheme that adjusts the regularisation/inflation parameter in the EKI update formula based on the evolution of the ensemble, without requiring external tuning parameters.
• Implement a data-driven early stopping criterion based on the convergence of the ensemble mean and covariance, preventing overfitting and reducing computational cost.
• Use a truncated Whittle-Matern (WM) level-set function to parameterise spatially heterogeneous physical properties, enabling accurate representation of sharp discontinuities and anomalies.
• Combine the WM level-set with additional WM fields to model spatial variability within distinct regions, enhancing flexibility in characterising complex material or tissue properties.
• Integrate the adaptive EKI framework with PDE-constrained optimisation to solve inverse problems in medical imaging and non-destructive testing.
• Validate the method using synthetic data from forward models simulating real-world measurement processes in NDT and medical imaging.

Experimental results

Research questions

• RQ1 Can an adaptive regularisation strategy in EKI eliminate the need for manual tuning of the inflation parameter while improving estimation accuracy?
• RQ2 How does the proposed early stopping criterion affect convergence speed and robustness compared to fixed-iteration schemes in EKI?
• RQ3 To what extent can the combination of truncated Whittle-Matern level-set functions and additional WM fields accurately represent heterogeneous and discontinuous physical properties in inverse problems?
• RQ4 Does the adaptive EKI framework outperform standard EKI in terms of computational efficiency and reconstruction accuracy for problems with sharp discontinuities?
• RQ5 Can the proposed method be effectively applied to real-world inverse problems in non-destructive testing and medical imaging with limited data and high noise?

Key findings

• The adaptive regularisation strategy successfully eliminates the need for tuning additional parameters, significantly improving the robustness of EKI in complex inverse problems.
• The method achieves accurate and stable reconstructions of heterogeneous physical properties with fewer iterations and smaller ensembles compared to standard EKI.
• The early stopping criterion prevents overfitting and reduces computational cost by terminating the algorithm once sufficient convergence is reached.
• The use of truncated Whittle-Matern level-set functions enables effective characterisation of sharp discontinuities, such as defects or diseased tissue, in the unknown field.
• Numerical experiments demonstrate that the proposed framework outperforms standard EKI in both reconstruction accuracy and computational efficiency across multiple test cases.
• The method is viable for practical applications in non-destructive testing and medical imaging, where data is limited and models are complex.