Ensemble Kalman Inversion for nonlinear problems: weights, consistency, and variance bounds

Abstract

Ensemble Kalman Inversion (EnKI) and Ensemble Square Root Filter (EnSRF) are popular sampling methods for obtaining a target posterior distribution. They can be seem as one step (the analysis step) in the data assimilation method Ensemble Kalman Filter. Despite their popularity, they are, however, not unbiased when the forward map is nonlinear. Important Sampling (IS), on the other hand, obtains the unbiased sampling at the expense of large variance of weights, leading to slow convergence of high moments. We propose WEnKI and WEnSRF, the weighted versions of EnKI and EnSRF in this paper. It follows the same gradient flow as that of EnKI/EnSRF with weight corrections. Compared to the classical methods, the new methods are unbiased, and compared with IS, the method has bounded weight variance. Both properties will be proved rigorously in this paper. We further discuss the stability of the underlying Fokker-Planck equation. This partially explains why EnKI, despite being inconsistent, performs well occasionally in nonlinear settings. Numerical evidence will be demonstrated at the end.

References in corpus (11)

• The Ensemble Kalman Filter for Inverse Problems
• A well-posedness theory in measures for some kinetic models of collective motion
• Parameterizations for Ensemble Kalman Inversion
• Deterministic Mean-field Ensemble Kalman Filtering
• Well Posedness and Convergence Analysis of the Ensemble Kalman Inversion
• A mollified Ensemble Kalman filter
• Long-time stability and accuracy of the ensemble Kalman-Bucy filter for fully observed processes and small measurement noise
• Ensemble Kalman Sampler: mean-field limit and convergence analysis
• On the continuous time limit of the Ensemble Kalman Filter
• Accelerating Langevin Sampling with Birth-death
• Continuous Limits for Constrained Ensemble Kalman Filter

Cited by in corpus (6)

• Ensemble Kalman Inversion: mean-field limit and convergence analysis
• Localization in Ensemble Kalman inversion
• Gradient flow structure and convergence analysis of the ensemble Kalman inversion for nonlinear forward models
• Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering
• Random Coordinate Underdamped Langevin Monte Carlo
• Sequential Kalman Tuning of the $t$-preconditioned Crank-Nicolson algorithm: efficient, adaptive and gradient-free inference for Bayesian inverse problems