Localization in Ensemble Kalman inversion

Tong, Xin; Morzfeld, Matthias

doi:10.48550/arxiv.2201.10821

Cited by 8 publications

(10 citation statements)

References 32 publications

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“…In this study, we limit the use of EKI and UKI to the calibration of dynamical models for which using an ensemble size J ∼ p is feasible. For models with a large number of parameters, localization or sampling error correction techniques can be used to maintain performance with J ≪ p members (Lee, 2021;Tong & Morzfeld, 2022), like in EnKF for data assimilation (Anderson, 2012). The update 18 also drives the ensemble toward consensus, in the sense that 𝐴𝐴 |Cov (𝜃𝜃𝑛𝑛, 𝑛𝑛) | → 0 as n → ∞; a popular method to control collapse speed is additive inflation (Anderson & Anderson, 1999;Tong & Morzfeld, 2022).…”

Section: Ensemble Kalman Inversion (Eki)mentioning

confidence: 99%

Training Physics‐Based Machine‐Learning Parameterizations With Gradient‐Free Ensemble Kalman Methods

Lopez‐Gomez

Christopoulos

Ervik

et al. 2022

J Adv Model Earth Syst

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Most machine learning applications in Earth system modeling currently rely on gradientbased supervised learning. This imposes stringent constraints on the nature of the data used for training (typically, residual time tendencies are needed), and it complicates learning about the interactions between machine-learned parameterizations and other components of an Earth system model. Approaching learning about process-based parameterizations as an inverse problem resolves many of these issues, since it allows parameterizations to be trained with partial observations or statistics that directly relate to quantities of interest in long-term climate projections. Here we demonstrate the effectiveness of Kalman inversion methods in treating learning about parameterizations as an inverse problem. We consider two different algorithms: unscented and ensemble Kalman inversion. Both methods involve highly parallelizable forward model evaluations, converge exponentially fast, and do not require gradient computations. In addition, unscented Kalman inversion provides a measure of parameter uncertainty. We illustrate how training parameterizations can be posed as a regularized inverse problem and solved by ensemble Kalman methods through the calibration of an eddy-diffusivity mass-flux scheme for subgrid-scale turbulence and convection, using data generated by large-eddy simulations. We find the algorithms amenable to batching strategies, robust to noise and model failures, and efficient in the calibration of hybrid parameterizations that can include empirical closures and neural networks.

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Section: Ensemble Kalman Inversion (Eki)mentioning

confidence: 99%

Training Physics‐Based Machine‐Learning Parameterizations With Gradient‐Free Ensemble Kalman Methods

Lopez‐Gomez

Christopoulos

Ervik

et al. 2022

J Adv Model Earth Syst

View full text Add to dashboard Cite

show abstract

“…In this study, we limit the use of EKI and UKI to the calibration of dynamical models for which J ∼ p is feasible. For models with a large number of parameters, localization techniques can be used to maintain performance with J p (Tong & Morzfeld, 2022).…”

Section: Ensemble Kalman Inversion (Eki)mentioning

confidence: 99%

Training physics-based machine-learning parameterizations with gradient-free ensemble Kalman methods

Lopez‐Gomez

Christopoulos

Ervik

et al. 2022

Preprint

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Most machine learning applications in Earth system modeling currently rely on gradientbased supervised learning. This imposes stringent constraints on the nature of the data used for training (typically, residual time tendencies are needed), and it complicates learning about the interactions between machine-learned parameterizations and other components of an Earth system model. Approaching learning about process-based parameterizations as an inverse problem resolves many of these issues, since it allows parameterizations to be trained with partial observations or statistics that directly relate to quantities of interest in long-term climate projections. Here we demonstrate the effectiveness of Kalman inversion methods in treating learning about parameterizations as an inverse problem. We consider two different algorithms: unscented and ensemble Kalman inversion. Both methods involve highly parallelizable forward model evaluations, converge exponentially fast, and do not require gradient computations. In addition, unscented Kalman inversion provides a measure of parameter uncertainty. How learning about parameterizations can be posed as an inverse problem and solved by ensemble Kalman methods is illustrated through the calibration of an eddy-diffusivity mass-flux scheme for subgridscale turbulence and convection, using data generated by large-eddy simulations. We find the algorithms amenable to batching strategies, robust to noise and model failures, and efficient in the calibration of hybrid parameterizations that can include empirical closures and neural networks.

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“…Related to this second theme, several works (e.g. [82,12,13,38,19,93,68]) set the analysis in a continuum limit; the idea is to view Kalman updates as occurring over an artificial discrete-time variable, and then take the time between updates to be infinitesimally small to formally derive differential equations for the evolution of the ensemble or its density. Large N asymptotics and continuum limits have resulted in new theoretical insights and practical advancements.…”

Section: Related Workmentioning

confidence: 99%

“…As discussed in [19], an entire suite of ensemble algorithms have been derived that differ in the choice of objective function and optimization scheme. In this subsection we introduce the Ensemble Kalman Inversion (EKI) algorithm [48] and a new localized implementation of EKI, which we call localized EKI (LEKI) following [93]. Both EKI and LEKI use an ensemble approximation of a Levenberg-Marquardt (LM) optimization scheme to minimize a data-misfit objective…”

Section: Ensemble Algorithms For Sequential Optimizationmentioning

confidence: 99%

“…when the underlying variables are further apart, the Schur-product operation zeroes out the corresponding elements of the estimator, and the rate at which this tapering occurs is controlled by the size of the length-scale. The localized EKI (LEKI), recently studied in [93], replaces both C pp and C up with their localized counterparts, M 1 • C pp and M 2 • C up , where M 1 and M 2 are localization matrices of appropriate dimension. Two important issues have, in our opinion, hindered the rigorous study of localized ensemble algorithms, and we highlight these next before moving on to introduce our localization framework.…”

Section: Localized Ensemble Kalman Inversion Updatementioning

confidence: 99%

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Non-Asymptotic Analysis of Ensemble Kalman Updates: Effective Dimension and Localization

Ghattas¹,

Sanz-Alonso²

2022

Preprint

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Many modern algorithms for inverse problems and data assimilation rely on ensemble Kalman updates to blend prior predictions with observed data. Ensemble Kalman methods often perform well with a small ensemble size, which is essential in applications where generating each particle is costly. This paper develops a non-asymptotic analysis of ensemble Kalman updates that rigorously explains why a small ensemble size suffices if the prior covariance has moderate effective dimension due to fast spectrum decay or approximate sparsity. We present our theory in a unified framework, comparing several implementations of ensemble Kalman updates that use perturbed observations, square root filtering, and localization. As part of our analysis, we develop new dimension-free covariance estimation bounds for approximately sparse matrices that may be of independent interest.

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Localization in Ensemble Kalman inversion

Cited by 8 publications

References 32 publications

Training Physics‐Based Machine‐Learning Parameterizations With Gradient‐Free Ensemble Kalman Methods

Training Physics‐Based Machine‐Learning Parameterizations With Gradient‐Free Ensemble Kalman Methods

Training physics-based machine-learning parameterizations with gradient-free ensemble Kalman methods

Non-Asymptotic Analysis of Ensemble Kalman Updates: Effective Dimension and Localization

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