On the incorporation of box-constraints for ensemble Kalman inversion

Neil K. Chada; Claudia Schillings; Simon Weissmann

doi:10.3934/fods.2019018

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2019, Volume 1, Issue 4: 433-456. Doi: 10.3934/fods.2019018

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On the incorporation of box-constraints for ensemble Kalman inversion

1.
Department of Statistics and Applied Probability, National University of Singapore, 119077, Singapore
2.
Institute of Mathematics, University of Mannheim, 68131 Mannheim, Germany

^* Corresponding author: Neil K. Chada
^* Corresponding author: Neil K. Chada

Published: December 2019

The first author is supported by a Singapore Ministry of Education Academic Research Funds Tier 2 grant [MOE2016-T2-2-135]. The third author is grateful to the DFG RTG1953 "Statistical Modeling of Complex Systems and Processes" for funding of this research

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Abstract

The Bayesian approach to inverse problems is widely used in practice to infer unknown parameters from noisy observations. In this framework, the ensemble Kalman inversion has been successfully applied for the quantification of uncertainties in various areas of applications. In recent years, a complete analysis of the method has been developed for linear inverse problems adopting an optimization viewpoint. However, many applications require the incorporation of additional constraints on the parameters, e.g. arising due to physical constraints. We propose a new variant of the ensemble Kalman inversion to include box constraints on the unknown parameters motivated by the theory of projected preconditioned gradient flows. Based on the continuous time limit of the constrained ensemble Kalman inversion, we discuss a complete convergence analysis for linear forward problems. We adopt techniques from filtering, such as variance inflation, which are crucial in order to improve the performance and establish a correct descent. These benefits are highlighted through a number of numerical examples on various inverse problems based on partial differential equations.

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Mathematics Subject Classification: Primary: 37C10, 49M15, 65M32; Secondary: 65N20.

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Figure 1. Varying contour lines of the function $ \Phi(x) $ defined in Example 2.2, with both the preconditioned descent direction in the unconstrained case and the projected preconditioned descent direction

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Figure 2. Transformed EnKF estimation in comparison to the EnKF estimation and the projected EnKF estimation. $ J = 5 $ particles have been simulated

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Figure 3. Ensemble spread in the transformed EnKF in comparison to the EnKF and the projected EnKF. $ J = 5 $ particles have been simulated

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Figure 4. KKT-Residuals and difference of the misfit functional and the global minimum in the transformed EnKF in comparison to the projected EnKF. $ J = 5 $ particles have been simulated

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Figure 5. Transformed EnKF estimation in comparison to the EnKF estimation and the projected EnKF estimation. $ J = 5 $ particles have been simulated

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Figure 6. Difference of the misfit functional and the global minimum in the transformed EnKF in comparison to the EnKF and the projected EnKF. $ J = 5 $ particles have been simulated

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Figure 7. Transformed EnKF parameter estimation in comparison to the EnKF estimation and the projected EnKF estimation. $ J = 5 $ particles have been simulated

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Figure 8. Transformed EnKF observation estimation in comparison to the EnKF estimation and the projected EnKF estimation. $ J = 5 $ particles have been simulated

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Figure 9. Ensemble spread in the transformed EnKF in comparison to the EnKF and the projected EnKF. $ J = 5 $ particles have been simulated

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Figure 10. Difference of the misfit functional and the global minimum in the transformed EnKF in comparison to the projected EnKF. $ J = 5 $ particles have been simulated

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