Analysis of the ensemble Kalman filter for inverse problems

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

A Stabilization of a Continuous Limit of the Ensemble Kalman Filter

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

Among the class of nonlinear particle filtering methods, the Ensemble Kalman Filter (EnKF) has gained recent attention for its use in solving inverse problems. We review the original method and discuss recent developments in particular in view of the limit for infinitely particles and extensions towards stability analysis and multi--objective optimization. We illustrate the performance of the method by using test inverse problems from the literature

Convergence Analysis of the Ensemble Kalman Filter for Inverse Problems: the Noisy Case

We present an analysis of the ensemble Kalman filter for inverse problems based on the continuous time limit of the algorithm. The analysis of the dynamical behaviour of the ensemble allows to establish well-posedness and convergence results for a fixed ensemble size. We will build on the results presented in [Schillings, Stuart 2017] and generalise them to the case of noisy observational data, in particular the influence of the noise on the convergence will be investigated, both theoretically and numerically

A strongly convergent numerical scheme from Ensemble Kalman inversion

The Ensemble Kalman methodology in an inverse problems setting can be viewed as an iterative scheme, which is a weakly tamed discretization scheme for a certain stochastic differential equation (SDE). Assuming a suitable approximation result, dynamical properties of the SDE can be rigorously pulled back via the discrete scheme to the original Ensemble Kalman inversion. The results of this paper make a step towards closing the gap of the missing approximation result by proving a strong convergence result in a simplified model of a scalar stochastic differential equation. We focus here on a toy model with similar properties than the one arising in the context of Ensemble Kalman filter. The proposed model can be interpreted as a single particle filter for a linear map and thus forms the basis for further analysis. The difficulty in the analysis arises from the formally derived limiting SDE with non-globally Lipschitz continuous nonlinearities both in the drift and in the diffusion. Here the standard Euler-Maruyama scheme might fail to provide a strongly convergent numerical scheme and taming is necessary. In contrast to the strong taming usually used, the method presented here provides a weaker form of taming. We present a strong convergence analysis by first proving convergence on a domain of high probability by using a cut-off or localisation, which then leads, combined with bounds on moments for both the SDE and the numerical scheme, by a bootstrapping argument to strong convergence