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

Kwiatkowski, Evan; Mandel, Jan

doi:10.1137/140965363

Cited by 55 publications

(69 citation statements)

References 26 publications

(35 reference statements)

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“…While the robust behavior of EnKFs has been demonstrated for many applications primarily arising from the geosciences, our theoretical understanding of their long-time stability and accuracy is still rather limited. Large sample size limits have been, for example, investigated in [GMT11,KM15] and it has been demonstrated that the EnKF converges to the classic Kalman filter for linear systems (1), linear observations (2), and Gaussian initial conditions. Using concepts from shadowing, [GTH13] showed that the EnKF is stable and accurate uniformly in time for hyperbolic dynamical systems provided the ensemble size is larger than the dimension of the chaotic attractor.…”

mentioning

confidence: 99%

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

Wiljes¹,

Reich²,

Stannat³

2018

SIAM J. Appl. Dyn. Syst.

141

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Abstract. 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.

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mentioning

confidence: 99%

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

Wiljes¹,

Reich²,

Stannat³

2018

SIAM J. Appl. Dyn. Syst.

141

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“…In addition to the EKI based on perturbed observations, we will also consider the ensemble square root filter (ESRF) applied to inverse problems. The ESRF [24,28,37] is a modification of the EnKF, but with the key difference of being deterministic as there is no inclusion of the perturbed observations. We emphasize our work will be primarily focused on the EnKF for inverse problems.…”

Section: Introductionmentioning

confidence: 99%

On the incorporation of box-constraints for ensemble Kalman inversion

Chada¹,

Schillings²,

Weissmann³

2019

Foundations of Data Science

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The Bayesian approach to inverse problems is widely used in practice to infer unknown parameters from noisy observations. In this framework, the ensemble Kalman inversion has been successfully applied for the quantification of uncertainties in various areas of applications. In recent years, a complete analysis of the method has been developed for linear inverse problems adopting an optimization viewpoint. However, many applications require the incorporation of additional constraints on the parameters, e.g. arising due to physical constraints. We propose a new variant of the ensemble Kalman inversion to include box constraints on the unknown parameters motivated by the theory of projected preconditioned gradient flows. Based on the continuous time limit of the constrained ensemble Kalman inversion, we discuss a complete convergence analysis for linear forward problems. We adopt techniques from filtering, such as variance inflation, which are crucial in order to improve the performance and establish a correct descent. These benefits are highlighted through a number of numerical examples on various inverse problems based on partial differential equations.

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“…The well-posedness of the EnKF and its accuracy using the variance inflation technique is studied in [12]. Related finitetime results on the convergence of the discrete-time square root EnKF appear in [13]. The analysis in [13] is simpler as the model is deterministic and the update formula exactly equals the Kalman filter update formula.…”

Section: Introductionmentioning

confidence: 99%

“…Related finitetime results on the convergence of the discrete-time square root EnKF appear in [13]. The analysis in [13] is simpler as the model is deterministic and the update formula exactly equals the Kalman filter update formula. The analysis for EnKBF and linear FPF is more recent.…”

Section: Introductionmentioning

confidence: 99%

Error Analysis of the Stochastic Linear Feedback Particle Filter

Taghvaei

Mehta

2018

2018 IEEE Conference on Decision and Control (CDC)

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This paper is concerned with the convergence and long-term stability analysis of the feedback particle filter (FPF) algorithm. The FPF is an interacting system of N particles where the interaction is designed such that the empirical distribution of the particles approximates the posterior distribution. It is known that in the mean-field limit (N = ∞), the distribution of the particles is equal to the posterior distribution. However little is known about the convergence to the mean-field limit. In this paper, we consider the FPF algorithm for the linear Gaussian setting. In this setting, the algorithm is similar to the ensemble Kalman-Bucy filter algorithm. Although these algorithms have been numerically evaluated and widely used in applications, their convergence and long-term stability analysis remains an active area of research. In this paper, we show that, (i) the mean-field limit is well-defined with a unique strong solution; (ii) the mean-field process is stable with respect to the initial condition; (iii) we provide conditions such that the finite-N system is long term stable and we obtain some mean-squared error estimates that are uniform in time.

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

Cited by 55 publications

References 26 publications

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

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

On the incorporation of box-constraints for ensemble Kalman inversion

Error Analysis of the Stochastic Linear Feedback Particle Filter

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