Multilevel Ensemble Kalman–Bucy Filters

Chada, Neil K.; Jasra, Ajay; Yu, Fangyuan

doi:10.1137/21m1423762

Cited by 7 publications

(17 citation statements)

References 38 publications

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“…As we have introduced the underlying concepts of MLMC, we now combine this with the EnKBF to result in the MLEnKBF. The discussion here was first presented in [6]. As we are considering a ML setting for the EnKBF we require two couplings, therefore we run the coupled particle systems of firstly, what we refer to as the multilevel VEnKBF (MLVEnKBF)…”

Section: Multilevel Ensemble Kalman-bucy Filtermentioning

confidence: 99%

“…The work of Chada et al [6] presents a ML analysis for the vanilla variant (F1), with a slight modification. The theory derived is presented for a simpler ideal i.i.d.…”

Section: Error-to-cost Rate Of Mlenkbfmentioning

confidence: 99%

“…However there has been a growing interest in the EnKBF for both theory and applications. In particular the work of Del Moral and coauthors have aimed to analyze various theoretical properties [3,13,14], while other directions have been on applications to non-linear filtering and uncertainty quantification [6,9,32,35,36].…”

Section: Introductionmentioning

confidence: 99%

“…This will be done for two of the unbiased estimators introduced in [31] which are the singleterm estimator and the coupled sum estimator. In order to assess the performance of the unbiased estimator, we compare this with the multilevel EnKBF (MLEnKBF) [6,32]. The multilevel approach [17,18] is a variance reduction approach for Monte Carlo which has seen particular interest in the filtering community [23,26].…”

Section: Introductionmentioning

confidence: 99%

“…This will be tested on a linear Ornstein-Uhlenbeck process, where we compare our unbiased estimators of the EnKBF to two particular variants of the MLEnKBF. To complement this work, we also derive MLMC rates, for the deterministic EnKBF, which is an extension of the work of [6], whose results were specific to the vanilla version. We will conduct this on two different dimensional examples, i.e.…”

Section: Introductionmentioning

confidence: 99%

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Unbiased Estimation of the Vanilla and Deterministic Ensemble Kalman-Bucy Filters

Álvarez¹,

Chada²,

Jasra³

2022

Preprint

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In this article we consider the development of an unbiased estimator for the ensemble Kalman-Bucy filter (EnKBF). The EnKBF is a continuous-time filtering methodology which can be viewed as a continuous-time analogue of the famous discrete-time ensemble Kalman filter. Our unbiased estimators will be motivated from recent work [Rhee & Glynn 2010, [31]] which introduces randomization as a means to produce unbiased and finite variance estimators. The randomization enters through both the level of discretization, and through the number of samples at each level. Our estimator will be specific to linear and Gaussian settings, where we know that the EnKBF is consistent, in the particle limit N → ∞, with the KBF. We highlight this for two particular variants of the EnKBF, i.e. the deterministic and vanilla variants, and demonstrate this on a linear Ornstein-Uhlenbeck process. We compare this with the EnKBF and the multilevel (MLEnKBF), for experiments with varying dimension size. We also provide a proof of the multilevel deterministic EnKBF, which provides a guideline for some of the unbiased methods.

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Section: Multilevel Ensemble Kalman-bucy Filtermentioning

confidence: 99%

“…The work of Chada et al [6] presents a ML analysis for the vanilla variant (F1), with a slight modification. The theory derived is presented for a simpler ideal i.i.d.…”

Section: Error-to-cost Rate Of Mlenkbfmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Unbiased Estimation of the Vanilla and Deterministic Ensemble Kalman-Bucy Filters

Álvarez¹,

Chada²,

Jasra³

2022

Preprint

Self Cite

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On the mathematical theory of ensemble (linear-Gaussian) Kalman–Bucy filtering

Bishop

Moral

2023

Math. Control Signals Syst.

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The purpose of this review is to present a comprehensive overview of the theory of ensemble Kalman–Bucy filtering for continuous-time, linear-Gaussian signal and observation 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 is signal tracking performance, and we contrast these results with those in classical studies of stability in Kalman–Bucy filtering. We also provide a novel (and negative) result proving that the bootstrap particle filter cannot track even the most basic unstable latent signal, in contrast with the ensemble Kalman filter (and the optimal filter). We provide intuition for how the main results extend to nonlinear signal models and comment on their consequence on some typical filter behaviours seen in practice, e.g. catastrophic divergence.

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