Continuous limits for constrained ensemble Kalman filter

Herty, Michaël; Visconti, Giuseppe

doi:10.1088/1361-6420/ab8bc5

Cited by 20 publications

(14 citation statements)

References 39 publications

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“…The dynamical system for the first and second moment is computed from (15). Using the linearity of the model we obtain (17)…”

Section: Stability Of the Moment Equationsmentioning

confidence: 99%

“…The update formula for each ensemble member is computed by imposing first order necessary optimality conditions to solve a regularized minimization problem, which aims for a compromise between the background estimate of the dynamics model and additional information provided by data model. A similar technique is used to derive the update formula for constrained inverse problems [2,17].…”

Section: Introductionmentioning

confidence: 99%

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A Stabilization of a Continuous Limit of the Ensemble Kalman Inversion

Armbruster¹,

Herty²,

Visconti³

2020

Preprint

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The ensemble Kalman filter belongs to the class of iterative particle filtering methods and can be used for solving control-to-observable inverse problems. In recent years several continuous limits in the number of iteration and particles have been performed in order to study properties of the method. In particular, a one-dimensional linear stability analysis reveals a possible instability of the solution provided by the continuous-time limit of the ensemble Kalman filter for inverse problems. In this work we address this issue by introducing a stabilization of the dynamics which leads to a method with globally asymptotically stable solutions. We illustrate the performance of the stabilized version of the ensemble Kalman filter by using test inverse problems from the literature and comparing it with the classical formulation of the method.

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“…The dynamical system for the first and second moment is computed from (15). Using the linearity of the model we obtain (17)…”

Section: Stability Of the Moment Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Stabilization of a Continuous Limit of the Ensemble Kalman Inversion

Armbruster¹,

Herty²,

Visconti³

2020

Preprint

Self Cite

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“…Earlier works are found in [34,16,29]. See also the numerical analysis and other follow up works in [7,8,22,41].…”

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confidence: 93%

Constrained Ensemble Langevin Monte Carlo

Ding¹,

Li²

2022

FoDS

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<p style='text-indent:20px;'>The classical Langevin Monte Carlo method looks for samples from a target distribution by descending the samples along the gradient of the target distribution. The method enjoys a fast convergence rate. However, the numerical cost is sometimes high because each iteration requires the computation of a gradient. One approach to eliminate the gradient computation is to employ the concept of "ensemble." A large number of particles are evolved together so the neighboring particles provide gradient information to each other. In this article, we discuss two algorithms that integrate the ensemble feature into LMC, and the associated properties.</p><p style='text-indent:20px;'>In particular, we find that if one directly surrogates the gradient using the ensemble approximation, the algorithm, termed Ensemble Langevin Monte Carlo, is unstable due to a high variance term. If the gradients are replaced by the ensemble approximations only in a constrained manner, to protect from the unstable points, the algorithm, termed Constrained Ensemble Langevin Monte Carlo, resembles the classical LMC up to an ensemble error but removes most of the gradient computation.</p>

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“…The recent popularity of CBO type methods stems, in particular, from their simple first order structure and the fact that, for high-dimensional, non convex, nonlinear unconstrained problems, numerical and analytical evidence of their performance has been reported (see [13,32] for machine learning applications). Related lines of research have considered mean-field descriptions of particle swarm optimization strategies [29,37] and nonlinear sampling techniques, like Ensemble Kalman filtering [25,35]. For an exhaustive list of references as well as a review of existing recent results, we refer to the recent survey articles [11,28,49].…”

Section: Introductionmentioning

confidence: 99%

Constrained consensus-based optimization

Borghi¹,

Herty²,

Pareschi³

2021

Preprint

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In this work we are interested in the construction of numerical methods for high dimensional constrained nonlinear optimization problems by particle-based gradient-free techniques. A consensus-based optimization (CBO) approach combined with suitable penalization techniques is introduced for this purpose. The method relies on a reformulation of the constrained minimization problem in an unconstrained problem for a penalty function and extends to the constrained settings the class of CBO methods. Exact penalization is employed and, since the optimal penalty parameter is unknown, an iterative strategy is proposed that successively updates the parameter based on the constrained violation. Using a mean-field description of the the many particle limit of the arising CBO dynamics, we are able to show convergence of the proposed method to the minimum for general nonlinear constrained problems. Properties of the new algorithm are analyzed. Several numerical examples, also in high dimensions, illustrate the theoretical findings and the good performance of the new numerical method.

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Continuous limits for constrained ensemble Kalman filter

Cited by 20 publications

References 39 publications

A Stabilization of a Continuous Limit of the Ensemble Kalman Inversion

A Stabilization of a Continuous Limit of the Ensemble Kalman Inversion

Constrained Ensemble Langevin Monte Carlo

Constrained consensus-based optimization

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