Estimate and control the correction term in Ensemble Kalman Inversion for nonlinear forward maps

Determine quantitative estimates for the correction term R(t,x;ρ) appearing in the PDE ∂tρ(t,x)=L[ρ(t,x)]+R(t,x;ρ)ρ(t,x) that underlies Ensemble Kalman Inversion when the forward map G is nonlinear, and develop a concrete control/weighting strategy for particles based on R(t,x;ρ) to ensure that the resulting ensemble accurately approximates the posterior density proportional to exp(−Φ(x;y)−|x−x0|^2_{Γ0}/2).

Background

Ensemble Kalman Inversion (EKI) uses an interacting particle system to approximate a PDE whose solution bridges the prior and posterior distributions over a pseudo-time horizon. For general nonlinear forward maps G, the correct PDE requires an extra correction term R(t,x;ρ); without accounting for this term, EKI may yield samples O(1) away from the target distribution.

The authors point out that this correction suggests particles should be reweighted according to R, but there is currently no rigorous framework for estimating or controlling this term in practice. Addressing this would clarify when EKI reliably approximates the posterior and how to implement the necessary weighting.