# Non-Asymptotic Analysis of Ensemble Kalman Updates: Effective Dimension and Localization

@article{Ghattas2022NonAsymptoticAO, title={Non-Asymptotic Analysis of Ensemble Kalman Updates: Effective Dimension and Localization}, author={Omar Al Ghattas and Daniel Sanz-Alonso}, journal={ArXiv}, year={2022}, volume={abs/2208.03246} }

Many modern algorithms for inverse problems and data assimilation rely on ensemble Kalman updates to blend prior predictions with observed data. Ensemble Kalman methods often perform well with a small ensemble size, which is essential in applications where generating each particle is costly. This paper develops a non-asymptotic analysis of ensemble Kalman updates that rigorously explains why a small ensemble size suﬃces if the prior covariance has moderate eﬀective dimension due to fast…

## References

SHOWING 1-10 OF 91 REFERENCES

### A localization technique for ensemble Kalman filters

- Environmental Science
- 2009

This article states an ordinary differential equation (ODE) with solutions that are equivalent to the Kalman filter update over a unit time interval, and forms a gradient system with the observations as a cost functional that should find useful application in the context of nonlinear observation operators and observations that arrive continuously in time.

### Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants

- Environmental Science
- 2007

### Well posedness and convergence analysis of the ensemble Kalman inversion

- MathematicsInverse Problems
- 2019

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.

### Convergence of the Square Root Ensemble Kalman Filter in the Large Ensemble Limit

- Environmental Science, MathematicsSIAM/ASA J. Uncertain. Quantification
- 2015

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.

### Performance of Ensemble Kalman Filters in Large Dimensions

- Environmental Science
- 2016

Contemporary data assimilation often involves more than a million prediction variables. Ensemble Kalman filters (EnKF) have been developed by geoscientists. They are successful indispensable tools in…

### An Ensemble Adjustment Kalman Filter for Data Assimilation

- Environmental Science
- 2001

Abstract A theory for estimating the probability distribution of the state of a model given a set of observations exists. This nonlinear filtering theory unifies the data assimilation and ensemble…

### Ensemble Square Root Filters

- Environmental ScienceStatistical Methods for Climate Scientists
- 2022

Abstract Ensemble data assimilation methods assimilate observations using state-space estimation methods and low-rank representations of forecast and analysis error covariances. A key element of such…

### Large sample asymptotics for the ensemble Kalman filter

- Environmental Science
- 2009

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 state…

### Understanding the Ensemble Kalman Filter

- Environmental Science, Computer Science
- 2016

The EnKF is successfully used in data-assimilation applications with tens of millions of dimensions and implicitly assumes a linear Gaussian state-space model, and has also turned out to be remarkably robust to deviations from these assumptions in many applications.

### Analysis of the Ensemble and Polynomial Chaos Kalman Filters in Bayesian Inverse Problems

- MathematicsSIAM/ASA J. Uncertain. Quantification
- 2015

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.