Ensemble Kalman Inversion for nonlinear problems: weights, consistency,
and variance bounds
Ensemble Kalman Inversion (EnKI), originally derived from Enseble Kalman
Filter, is a popular sampling method for obtaining a target posterior
distribution. It is, however, inconsistent when the forward map is nonlinear.
Important Sampling (IS), on the other hand, ensures consistency at the expense
of large variance of weights, leading to slow convergence of high moments.
We propose a WEnKI, a weighted version of EnKI in this paper. It follows the
same gradient flow as that of EnKI with a weight correction. Compared to EnKI,
the new method is consistent, 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