A Conjugate Gradient Formulation of the EnKF Algorithm
Abstract
Ensemble Kalman Filter (EnKF) based data assimilation algorithms synthesize predictive numerical forecast models with accumulated data as time evolves and account for model uncertainty and noisy measurements. The computational cost of these algorithms can be expensive, in particular for highly dimensional dynamical systems. Often, EnKF based algorithms have traded accuracy for reduced computational cost. In this paper, we present a novel parallelizable Conjugate Gradient-based Ensemble Kalman Filter (CGD-EnKF) algorithm that maintains comparable computational cost to efficient algorithms while realizing better state estimation accuracy in select cases. Here, we established the new approach by reformulating a matrix inverse calculation with a classical Conjugate Gradient (CGD) method. In addition, we discuss the upper error bound under CGD, error convergence to the classical EnKF result, and the computational complexity of the algorithm. We also showcase the CGD-EnKF-Reduced algorithm that is shown to be further computationally efficient for highly dimensional dynamical systems under small ensemble formulation. Numerical examples demonstrate the performance of our proposed algorithms and analytical properties, highlighting their comparability and advantages with respect to some benchmark EnKF algorithms.
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BibTeX
@misc{lee2026conjugate,
title = {{A Conjugate Gradient Formulation of the EnKF Algorithm}},
author = {Sanghyun Lee and Zhengqi Liu and Jonathan Valyou and Ludmil Zikatanov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.19224},
eprint = {2606.19224},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.19224},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Conjugate Gradient Formulation of the EnKF Algorithm
AU - Sanghyun Lee
AU - Zhengqi Liu
AU - Jonathan Valyou
AU - Ludmil Zikatanov
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2606.19224
UR - https://arxiv.org/abs/2606.19224
AB - Ensemble Kalman Filter (EnKF) based data assimilation algorithms synthesize predictive numerical forecast models with accumulated data as time evolves and account for model uncertainty and noisy measurements. The computational cost of these algorithms can be expensive, in particular for highly dimensional dynamical systems. Often, EnKF based algorithms have traded accuracy for reduced computational cost. In this paper, we present a novel parallelizable Conjugate Gradient-based Ensemble Kalman Filter (CGD-EnKF) algorithm that maintains comparable computational cost to efficient algorithms while realizing better state estimation accuracy in select cases. Here, we established the new approach by reformulating a matrix inverse calculation with a classical Conjugate Gradient (CGD) method. In addition, we discuss the upper error bound under CGD, error convergence to the classical EnKF result, and the computational complexity of the algorithm. We also showcase the CGD-EnKF-Reduced algorithm that is shown to be further computationally efficient for highly dimensional dynamical systems under small ensemble formulation. Numerical examples demonstrate the performance of our proposed algorithms and analytical properties, highlighting their comparability and advantages with respect to some benchmark EnKF algorithms.
ER -