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

Ernst, Oliver G.; Sprungk, Björn; Starkloff, Hans-Jörg

doi:10.1137/140981319

Cited by 72 publications

(89 citation statements)

References 32 publications

(54 reference statements)

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“…The analysis of the method was also studied in the large ensemble limit, see e.g. [21,34,37,38]. However, to the best of our knowledge, the derivation of a kinetic equation that holds in the limit of a large number of ensembles has not yet been proposed.…”

Section: Mean-field Limit Of the Ensemble Kalman Filtermentioning

confidence: 99%

“…For further details concerning Bayesian inversion, e.g. the modeling of the unknown prior distributions and other choices of estimators, see [4,9,15,21] and references therein.…”

Section: Introductionmentioning

confidence: 99%

“…In the cited references the ensemble size was fixed and, due to the possible associated high computational cost, limited to a small number of ensembles. The analysis of the method for a large ensemble size limit has been investigated in [16,17,21,37]. However, to the best of our knowledge, an evolution equation for the probability distribution of the unknown control has not been derived.…”

Section: Introductionmentioning

confidence: 99%

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Kinetic methods for inverse problems

Herty¹,

Visconti²

2019

Kinetic &Amp; Related Models

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The Ensemble Kalman Filter method can be used as an iterative numerical scheme for parameter identification or nonlinear filtering problems. We study the limit of infinitely large ensemble size and derive the corresponding mean-field limit of the ensemble method. The solution of the inverse problem is provided by the expected value of the distribution of the ensembles and the kinetic equation allows, in simple cases, to analyze stability of these solutions. Further, we present a slight but stable modification of the method which leads to a Fokker-Planck-type kinetic equation. The kinetic methods proposed here are able to solve the problem with a reduced computational complexity in the limit of a large ensemble size. We illustrate the properties and the ability of the kinetic model to provide solution to inverse problems by using examples from the literature.Mathematics Subject Classification (2010) 35Q84, 65N21, 93E11, 65N75

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Section: Mean-field Limit Of the Ensemble Kalman Filtermentioning

confidence: 99%

“…For further details concerning Bayesian inversion, e.g. the modeling of the unknown prior distributions and other choices of estimators, see [4,9,15,21] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Kinetic methods for inverse problems

Herty¹,

Visconti²

2019

Kinetic &Amp; Related Models

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“…where G is again the finite difference discretization of the continuous linear operator defining the elliptic PDE (13). By using (36) we finally look at…”

Section: Numerical Experimentsmentioning

confidence: 99%

Continuous limits for constrained ensemble Kalman filter

Herty

Visconti

2020

Inverse Problems

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The Ensemble Kalman Filter method can be used as an iterative particle numerical scheme for state dynamics estimation and control-to-observable identification problems. In applications it may be required to enforce the solution to satisfy equality constraints on the control space. In this work we deal with this problem from a constrained optimization point of view, deriving corresponding optimality conditions. Continuous limits, in time and in the number of particles, allows us to study properties of the method. We illustrate the performance of the method by using test inverse problems from the literature.

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“…Subsequently, Dashti and Stuart [9] have reduced these assumptions significantly. Finally, we mention Ernst et al [14], who have discussed uniform and Hölder continuity of posterior measures with respect to data, and give sufficient assumptions in this setting. We refer to these as (Hölder, Hellinger) and (uniform, Hellinger) well-posedness, respectively.…”

mentioning

confidence: 99%

On the Well-posedness of Bayesian Inverse Problems

Latz¹

2020

SIAM/ASA J. Uncertainty Quantification

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The subject of this article is the introduction of a new concept of well-posedness of Bayesian inverse problems. The conventional concept of (Lipschitz, Hellinger) well-posedness in [Stuart 2010, Acta Numerica 19, pp. 451-559] is difficult to verify in practice and may be inappropriate in some contexts. Our concept simply replaces the Lipschitz continuity of the posterior measure in the Hellinger distance by continuity in an appropriate distance between probability measures. Aside from the Hellinger distance, we investigate well-posedness with respect to weak convergence, the total variation distance, the Wasserstein distance, and also the Kullback-Leibler divergence. We demonstrate that the weakening to continuity is tolerable and that the generalisation to other distances is important. The main results of this article are proofs of wellposedness with respect to some of the aforementioned distances for large classes of Bayesian inverse problems. Here, little or no information about the underlying model is necessary; making these results particularly interesting for practitioners using black-box models. We illustrate our findings with numerical examples motivated from machine learning and image processing.

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

Cited by 72 publications

References 32 publications

Kinetic methods for inverse problems

Kinetic methods for inverse problems

Continuous limits for constrained ensemble Kalman filter

On the Well-posedness of Bayesian Inverse Problems

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