On the incorporation of box-constraints for ensemble Kalman inversion

Chada, Neil K.; Schillings, Claudia; Weissmann, Simon

doi:10.3934/fods.2019018

Cited by 22 publications

(25 citation statements)

References 30 publications

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“…There are many recent works that carefully describe the mathematics of EKI [5,10], that suggest improvements [11,12,13,14], and that explain how EKI can be used in machine learning [15]. An extension of EKI, such that the EKI ensemble is distributed according to a Bayesian posterior distribution, is discussed in [16] and is called the ensemble Kalman sampler.…”

Section: Introductionmentioning

confidence: 99%

Localization in Ensemble Kalman inversion

Tong¹,

Morzfeld²

2022

Preprint

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Ensemble Kalman inversion (EKI) is a technique for the numerical solution of inverse problems. A great advantage of the EKI's ensemble approach is that derivatives are not required in its implementation. But theoretically speaking, EKI's ensemble size needs to surpass the dimension of the problem. This is because of EKI's "subspace property", i.e., that the EKI solution is a linear combination of the initial ensemble it starts off with. We show that the ensemble can break out of this initial subspace when "localization" is applied. In essence, localization enforces an assumed correlation structure onto the problem, and is heavily used in ensemble Kalman filtering and data assimilation. We describe and analyze how to apply localization to the EKI, and how localization helps the EKI ensemble break out of the initial subspace. Specifically, we show that the localized EKI (LEKI) ensemble will collapse to a single point (as intended) and that the LEKI ensemble mean will converge to the global optimum at a sublinear rate. Under strict assumptions on the localization procedure and observation process, we further show that the data misfit decays uniformly. We illustrate our ideas and theoretical developments with numerical examples with simplified toy problems, a Lorenz model, and an inversion of electromagnetic data, where some of our mathematical assumptions may only be approximately valid.

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Section: Introductionmentioning

confidence: 99%

Localization in Ensemble Kalman inversion

Tong¹,

Morzfeld²

2022

Preprint

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“…A constrained data assimilation problem can be tackled from different perspectives, which leads to a variety of constrained assimilation algorithms [2,34]. For instance, certain constrained assimilation algorithms are designed to handle a given type of constraints, e.g., equality constraints [19,35] or inequality constraints [6,32,33,40], or a mixture of both equality and inequality constraints [1,14,18].…”

Section: Introductionmentioning

confidence: 99%

Data assimilation with soft constraints (DASC) through a generalized iterative ensemble smoother

Luo

Cruz

2022

Comput Geosci

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This work investigates an ensemble-based workflow to simultaneously handle generic, nonlinear equality and inequality constraints in reservoir data assimilation problems. The proposed workflow is built upon a recently proposed umbrella algorithm, called the generalized iterative ensemble smoother (GIES), and inherits the benefits of ensemble-based data assimilation algorithms in geoscience applications. Unlike the traditional ensemble assimilation algorithms, the proposed workflow admits cost functions beyond the form of nonlinear-least-squares, and has the potential to develop an infinite number of constrained assimilation algorithms. In the proposed workflow, we treat data assimilation with constraints as a constrained optimization problem. Instead of relying on a general-purpose numerical optimization algorithm to solve the constrained optimization problem, we derive an (approximate) closed form to iteratively update model variables, but without the need to explicitly linearize the constraint systems. The established model update formula bears similarities to that of an iterative ensemble smoother (IES). Therefore, in terms of theoretical analysis, it becomes relatively easy to transit from an ordinary IES to the proposed constrained assimilation algorithms, and in terms of practical implementation, it is also relatively straightforward to implement the proposed workflow for users who are familiar with the IES, or other conventional ensemble data assimilation algorithms like the ensemble Kalman filter (EnKF). Apart from the aforementioned features, we also develop efficient methods to handle two noticed issues that would be of practical importance for ensemble-based constrained assimilation algorithms. These issues include localization in the presence of constraints, and the (possible) high dimensionality induced by the constraint systems. We use one 2D and one 3D case studies to demonstrate the performance of the proposed workflow. In particular, the 3D example contains experiment settings close to those of real field case studies. In both case studies, the proposed workflow achieves better data assimilation performance in comparison to the choice of using an original IES algorithm. As such, the proposed workflow has the potential to further improve the efficacy of ensemble-based data assimilation in practical reservoir data assimilation problems.

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“…The method proposed leverages the fact that the update step in the usual ensemble Kalman method can be formulated as an optimization problem, in which linear constraints can be integrated. In [15], a variant of EKI incorporating a projection step is developed for the specific case of box constraints, and the continuous time limit of the method is studied. See also [28], where the continuous-time and mean field limits of the method proposed in [2] are studied.…”

Section: Introductionmentioning

confidence: 99%

Consensus-based Optimization and Ensemble Kalman Inversion for Global Optimization Problems with Constraints

Carrillo¹,

Totzeck²,

Vaes³

2021

Preprint

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We introduce a practical method for incorporating equality and inequality constraints in global optimization methods based on stochastic interacting particle systems, specifically consensusbased optimization (CBO) and ensemble Kalman inversion (EKI). Unlike other approaches in the literature, the method we propose does not constrain the dynamics to the feasible region of the state space at all times; the particles evolve in the full space, but are attracted towards the feasible set by means of a penalization term added to the objective function and, in the case of CBO, an additional relaxation drift. We study the properties of the method through the associated mean-field Fokker-Planck equation and demonstrate its performance in numerical experiments on several test problems.

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On the incorporation of box-constraints for ensemble Kalman inversion

Cited by 22 publications

References 30 publications

Localization in Ensemble Kalman inversion

Localization in Ensemble Kalman inversion

Data assimilation with soft constraints (DASC) through a generalized iterative ensemble smoother

Consensus-based Optimization and Ensemble Kalman Inversion for Global Optimization Problems with Constraints

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