221 research outputs found
Long-time stability and accuracy of the ensemble Kalman--Bucy filter for fully observed processes and small measurement noise
The ensemble Kalman filter has become a popular data assimilation technique in the geosciences. However, little is known theoretically about its long term stability and accuracy. In this paper, we investigate the behavior of an ensemble Kalman--Bucy filter applied to continuous-time filtering problems. We derive mean field limiting equations as the ensemble size goes to infinity as well as uniform-in-time accuracy and stability results for finite ensemble sizes. The later results require that the process is fully observed and that the measurement noise is small. We also demonstrate that our ensemble Kalman--Bucy filter is consistent with the classic Kalman--Bucy filter for linear systems and Gaussian processes. We finally verify our theoretical findings for the Lorenz-63 system
Convergence Analysis of Ensemble Kalman Inversion: The Linear, Noisy Case
We present an analysis of ensemble Kalman inversion, based on the continuous
time limit of the algorithm. The analysis of the dynamical behaviour of the
ensemble allows us to establish well-posedness and convergence results for a
fixed ensemble size. We will build on the results presented in [26] 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. We focus on linear inverse problems where a very
complete theoretical analysis is possible
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
On the Mathematical Theory of Ensemble (Linear-Gaussian) Kalman-Bucy Filtering
The purpose of this review is to present a comprehensive overview of the
theory of ensemble Kalman-Bucy filtering for linear-Gaussian signal models. We
present a system of equations that describe the flow of individual particles
and the flow of the sample covariance and the sample mean in continuous-time
ensemble filtering. We consider these equations and their characteristics in a
number of popular ensemble Kalman filtering variants. Given these equations, we
study their asymptotic convergence to the optimal Bayesian filter. We also
study in detail some non-asymptotic time-uniform fluctuation, stability, and
contraction results on the sample covariance and sample mean (or sample error
track). We focus on testable signal/observation model conditions, and we
accommodate fully unstable (latent) signal models. We discuss the relevance and
importance of these results in characterising the filter's behaviour, e.g. it's
signal tracking performance, and we contrast these results with those in
classical studies of stability in Kalman-Bucy filtering. We provide intuition
for how these results extend to nonlinear signal models and comment on their
consequence on some typical filter behaviours seen in practice, e.g.
catastrophic divergence
Well Posedness and Convergence Analysis of the Ensemble Kalman Inversion
The ensemble Kalman inversion is widely used in practice to estimate unknown
parameters from noisy measurement data. Its low computational costs,
straightforward implementation, and non-intrusive nature makes the method
appealing in various areas of application. We present a complete analysis of
the ensemble Kalman inversion with perturbed observations for a fixed ensemble
size when applied to linear inverse problems. The well-posedness and
convergence results are based on the continuous time scaling limits of the
method. The resulting coupled system of stochastic differential equations
allows to derive estimates on the long-time behaviour and provides insights
into the convergence properties of the ensemble Kalman inversion. We view the
method as a derivative free optimization method for the least-squares misfit
functional, which opens up the perspective to use the method in various areas
of applications such as imaging, groundwater flow problems, biological problems
as well as in the context of the training of neural networks
- …