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

<p style='text-indent:20px;'>Ensemble Kalman Inversion (EnKI) [<xref ref-type="bibr" rid="b23">23</xref>] and Ensemble Square Root Filter (EnSRF) rid="b36">36</xref>] 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 rid="b17">17</xref>,<xref rid="b3">3</xref>]. Despite their popularity, they are, however, not unbiased when forward map is nonlinear rid="b12">12</xref>,<xref rid="b16">16</xref>,<xref rid="b25">25</xref>]. Important Sampling (IS), on other hand, obtains at expense of large variance weights, leading to slow convergence high moments.</p><p style='text-indent:20px;'>We propose WEnKI WEnSRF, weighted versions EnKI EnSRF this paper. It follows same gradient flow that EnKI/EnSRF with weight corrections. Compared classical methods, new unbiased, compared IS, has bounded variance. Both properties will proved rigorously We further discuss stability underlying Fokker-Planck equation. This partially explains why EnKI, despite being inconsistent, performs well occasionally settings. Numerical evidence demonstrated end.</p>