Corpus ID: 236635557

Continuous time limit of the stochastic ensemble Kalman inversion: Strong convergence analysis

@article{Blmker2021ContinuousTL,
  title={Continuous time limit of the stochastic ensemble Kalman inversion: Strong convergence analysis},
  author={Dirk Bl{\"o}mker and Claudia Schillings and Philipp Wacker and Simon Weissmann},
  journal={ArXiv},
  year={2021},
  volume={abs/2107.14508}
}
The Ensemble Kalman inversion (EKI) method is a method for the estimation of unknown parameters in the context of (Bayesian) inverse problems. The method approximates the underlying measure by an ensemble of particles and iteratively applies the ensemble Kalman update to evolve (the approximation of the) prior into the posterior measure. For the convergence analysis of the EKI it is common practice to derive a continuous version, replacing the iteration with a stochastic differential equation… Expand

Figures from this paper

Adaptive Tikhonov strategies for stochastic ensemble Kalman inversion
Ensemble Kalman inversion (EKI) is a derivative-free optimizer aimed at solving inverse problems, taking motivation from the celebrated ensemble Kalman filter. The purpose of this article is toExpand

References

SHOWING 1-10 OF 57 REFERENCES
On the continuous time limit of the ensemble Kalman filter
TLDR
The original Ensemble Kalman Filter algorithm proposed by [1] as well as a recent variant [2] to the respective discretizations are applied and it is shown that in the limit of decreasing stepsize the filter equations converge to an ensemble of interacting (stochastic) differential equations in the ensemble-mean-square sense. Expand
Well posedness and convergence analysis of the ensemble Kalman inversion
TLDR
This work views the ensemble Kalman inversion 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. Expand
Analysis of the Ensemble Kalman Filter for Inverse Problems
TLDR
The goal of this paper is to analyze the EnKF when applied to inverse problems with fixed ensemble size, and to demonstrate that the conclusions of the analysis extend beyond the linear inverse problem setting. Expand
A Stabilization of a Continuous Limit of the Ensemble Kalman Filter
TLDR
This work introduces a stabilization of the dynamics which leads to a method with globally asymptotically stable solutions of the ensemble Kalman filter for inverse problems. Expand
Convergence analysis of ensemble Kalman inversion: the linear, noisy case
TLDR
The analysis of the dynamical behaviour of the ensemble allows us to establish well-posedness and convergence results for a fixed ensemble size and focuses on linear inverse problems where a very complete theoretical analysis is possible. Expand
Ensemble Kalman methods for inverse problems
TLDR
It is demonstrated that the ensemble Kalman method for inverse problems provides a derivative-free optimization method with comparable accuracy to that achieved by traditional least-squares approaches, and that the accuracy is of the same order of magnitude as that achieve by the best approximation. Expand
Convergence of the Square Root Ensemble Kalman Filter in the Large Ensemble Limit
TLDR
It is shown that at every time index, as the number of ensemble members increases to infinity, the mean and covariance of an unbiased ensemble square root filter converge to those of the Kalman filter, in the case a linear model and an initial distribution of which all moments exist. Expand
Deterministic Mean-Field Ensemble Kalman Filtering
TLDR
A density-based deterministic approximation of the mean-field limit EnKF (DMFEnKF) is proposed, consisting of a PDE solver and a quadrature rule, which is therefore asymptotically superior to standard EnkF when the dimension $d<2\kappa$. Expand
Analysis of the Ensemble and Polynomial Chaos Kalman Filters in Bayesian Inverse Problems
TLDR
It is proved that, in the limit of large ensemble or high polynomial degree, both Kalman filters yield approximations which converge to a well-defined random variable termed the analysis random variable, and it is shown that this analysis variable is more closely related to a specific linear Bayes estimator than to the solution of the associated Bayesian inverse problem given by the posterior measure. Expand
Large sample asymptotics for the ensemble Kalman filter
The ensemble Kalman filter (EnKF) has been proposed as a Monte Carlo, derivative-free, alternative to the extended Kalman filter, and is now widely used in sequential data assimilation, where stateExpand
...
1
2
3
4
5
...