A Stabilization of a Continuous Limit of the Ensemble Kalman Inversion

Abstract

The ensemble Kalman filter belongs to the class of iterative particle filtering methods and can be used for solving control-to-observable inverse problems. In recent years several continuous limits in the number of iteration and particles have been performed in order to study properties of the method. In particular, a one-dimensional linear stability analysis reveals a possible instability of the solution provided by the continuous-time limit of the ensemble Kalman filter for inverse problems. In this work we address this issue by introducing a stabilization of the dynamics which leads to a method with globally asymptotically stable solutions. We illustrate the performance of the stabilized version of the ensemble Kalman filter by using test inverse problems from the literature and comparing it with the classical formulation of the method.

Recent Trends on Nonlinear Filtering for Inverse Problems

“…Note that within this limit the noise is scaled with 1 ∆t which allows for the continuous time limit. Further, the term D G vanishes leading to possibly unstable dynamics [27,5].…”

“…For the rigorous mean-field derivation and analysis of the EKI we refer to [10,16]. Equation ( 9) is a nonlinear transport equation arising from non-linear gradient flow interactions and in [10,5] it is observed that the counterpart of (8) holds at the kinetic level. In fact, for…”

“…In the continuous-time limit the term D G present in the discrete formulation vanish due to scaling. This consideration inspired [5], where a stability analysis of the moment system of the time-continuous EKI ( 7) is performed. Therein, it has been established that the system has infinitely many non-hyperbolic Bogdanov-Takens equilibria leading to several undesirable consequences.…”

“…These considerations led to a modified formulation of the method is globally asymptotically stable by introducing the regularization term R to the dynamics. More precisely, given Σ ∈ R d×d symmetric and full-rank, in particular positive definite, in [5] it is proposed to consider the following general discrete dynamics for each ensemble member j = 1, . .…”

Complete Deterministic Dynamics and Spectral Decomposition of the Linear Ensemble Kalman Inversion

“…Hence the evolution of the whole ensemble and the sample means can be seen as minimizing movement discretization of the gradient flow of the functional u → 1 2 Au − ỹ 2 Γ with respect to the varying norm • 2 C(un) , see for example the discussion in [1] and also [25].…”