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

Abstract

<p style='text-indent:20px;'>Ensemble Kalman Inversion (EnKI) [<xref ref-type="bibr" rid="b23">23</xref>] and Ensemble Square Root Filter (EnSRF) [<xref ref-type="bibr" 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 Ensemble Kalman Filter [<xref ref-type="bibr" rid="b17">17</xref>,<xref ref-type="bibr" rid="b3">3</xref>]. Despite their popularity, they are, however, not unbiased when the forward map is nonlinear [<xref ref-type="bibr" rid="b12">12</xref>,<xref ref-type="bibr" rid="b16">16</xref>,<xref ref-type="bibr" rid="b25">25</xref>]. 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.</p><p style='text-indent:20px;'>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.</p>