Analysis of the ensemble Kalman filter for inverse problems

TL;DR

This paper analyzes the ensemble Kalman filter's behavior in inverse problems with fixed ensemble size, deriving a continuous-time limit and studying its long-term dynamics, primarily for linear problems.

Contribution

It provides the first rigorous analysis of EnKF for inverse problems at fixed ensemble size, including a continuous-time limit and insights into its long-term behavior.

Findings

Continuous-time limit corresponds to gradient flows for data misfit.

Analysis primarily for linear inverse problems, extended numerically to nonlinear cases.

Numerical experiments show benefits of methodological extensions.

Abstract

The ensemble Kalman filter (EnKF) is a widely used methodology for state estimation in partial, noisily observed dynamical systems, and for parameter estimation in inverse problems. Despite its widespread use in the geophysical sciences, and its gradual adoption in many other areas of application, analysis of the method is in its infancy. Furthermore, much of the existing analysis deals with the large ensemble limit, far from the regime in which the method is typically used. The goal of this paper is to analyze the method when applied to inverse problems with fixed ensemble size. A continuous-time limit is derived and the long-time behavior of the resulting dynamical system is studied. Most of the rigorous analysis is confined to the linear forward problem, where we demonstrate that the continuous time limit of the EnKF corresponds to a set of gradient flows for the data misfit in each ensemble member, coupled through a common pre-conditioner which is the empirical covariance matrix of the ensemble. Numerical results demonstrate that the conclusions of the analysis extend beyond the linear inverse problem setting. Numerical experiments are also given which demonstrate the benefits of various extensions of the basic methodology.