On the continuous time limit of ensemble square root filters

@article{Lange2021OnTC,
  title={On the continuous time limit of ensemble square root filters},
  author={Theresa Lange and Wilhelm Stannat},
  journal={Communications in Mathematical Sciences},
  year={2021}
}
  • Theresa Lange, W. Stannat
  • Published 28 October 2019
  • Mathematics, Environmental Science
  • Communications in Mathematical Sciences
We provide a continuous time limit analysis for the class of Ensemble Square Root Filter algorithms with deterministic model perturbations. In the particular linear case, we specify general conditions on the model perturbations implying convergence of the empirical mean and covariance matrix towards their respective counterparts of the Kalman-Bucy Filter. As a second main result we identify additional assumptions for the convergence of the whole ensemble towards solutions of the Ensemble Kalman… 

Gradient flow structure and convergence analysis of the ensemble Kalman inversion for nonlinear forward models

TLDR
The effect of applying a sample covariance as preconditioning matrix is discussed and the gradient flow structure of the EKI is quantified by controlling the approximation error through the spread in the particle system.

Convergence Acceleration of Ensemble Kalman Inversion in Nonlinear Settings

TLDR
This article investigates EKI convergence in general nonlinear settings and proves that EKI can hit critical points with finite steps in non-convex settings and converges to the global minimizer polynomially fast if the loss function is strongly convex.

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

TLDR
This paper proves convergence in probability for the former, and convergence in moments for the latter of the stochastic EKI iteration converges to paths of the continuous-time Stochastic differential equation by considering both the nonlinear and linear setting.

Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering

TLDR
A McKean-Vlasov equation that contains the data stream as a common driving rough path is studied, establishing propagation of chaos for the associated interacting particle system, which is suggestive of a numerical scheme that can be viewed as an extension of the ensemble Kalman filter to a rough-path framework.

A theoretical analysis of one-dimensional discrete generation ensemble Kalman particle filters

Despite the widespread usage of discrete generation Ensemble Kalman particle filtering methodology to solve nonlinear and high dimensional filtering and inverse problems, little is known about their

Derivation of ensemble Kalman–Bucy filters with unbounded nonlinear coefficients

We provide a rigorous derivation of the ensemble Kalman–Bucy filter as well as the ensemble transform Kalman–Bucy filter in case of nonlinear, unbounded model and observation operators. We identify

Mean field limit of Ensemble Square Root filters - discrete and continuous time

Consider the class of Ensemble Square Root filtering algorithms for the numerical approximation of the posterior distribution of nonlinear Markovian signals partially observed with linear

Thermophysical modelling and parameter estimation of small solar system bodies via data assimilation

TLDR
A standard sequential data assimilation method, the Ensemble Square Root Filter, is introduced to thermophysical modelling of asteroid surfaces and, for the first time, correlations and uncertainties of all free model parameters are incorporated in the estimation procedure which is more than 5000 times more efficient than a comparable parameter sweep.

Analysis of a localised nonlinear Ensemble Kalman Bucy Filter with complete and accurate observations

TLDR
This article proposes and investigates a localized Ensemble Kalman Bucy Filter (l-EnKBF) for nonlinear models with short-range interactions and derives dimension-independent and component-wise error bounds and shows the long time path-wiseerror only has logarithmic dependence on the time range.

References

SHOWING 1-10 OF 27 REFERENCES

A New Approach to Linear Filtering and Prediction Problems

The clssical filleting and prediclion problem is re-examined using the Bode-Shannon representation of random processes and the ?stat-tran-sition? method of analysis of dynamic systems. New result

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.

Ensemble Square Root Filters

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

Long-Time Stability and Accuracy of the Ensemble Kalman-Bucy Filter for Fully Observed Processes and Small Measurement Noise

TLDR
This paper investigates the behavior of an ensemble Kalman-Bucy filter applied to continuous-time filtering problems and derives 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.

A deterministic formulation of the ensemble Kalman filter : an alternative to ensemble square root filters

The use of perturbed observations in the traditional ensemble Kalman filter (EnKF) results in a suboptimal filter behaviour, particularly for small ensembles. In this work, we propose a simple

New Results in Liner Filtering and Prediction Theory. Transactions of the ASME

  • Journal of Basic Engineering,
  • 1961

Mean field limit of Ensemble Square Root filters - discrete and continuous time

Consider the class of Ensemble Square Root filtering algorithms for the numerical approximation of the posterior distribution of nonlinear Markovian signals partially observed with linear

Numerical Solution of Stochastic Differential Equations

This paper provides an introduction to the main concepts and techniques necessary for someone who wishes to carryout numerical experiments involving Stochastic Differential Equation (SDEs). As SDEs

An inequality for trace ideals

We prove an inequality for trace ideals which relates the difference of two positive operators to the difference of their square roots. Inequalities involving operator-monotone functions more general