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Title: Multilevel ensemble Kalman filtering

Abstract

This study embeds a multilevel Monte Carlo sampling strategy into the Monte Carlo step of the ensemble Kalman filter (EnKF) in the setting of finite dimensional signal evolution and noisy discrete-time observations. The signal dynamics is assumed to be governed by a stochastic differential equation (SDE), and a hierarchy of time grids is introduced for multilevel numerical integration of that SDE. Finally, the resulting multilevel EnKF is proved to asymptotically outperform EnKF in terms of computational cost versus approximation accuracy. The theoretical results are illustrated numerically.

Authors:
 [1];  [2];  [3]
  1. Univ. of Oslo (Norway)
  2. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
  3. King Abdullah University of Science and Technology (KAUST), Thuwal (Saudi Arabia)
Publication Date:
Research Org.:
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Sponsoring Org.:
USDOE Laboratory Directed Research and Development (LDRD) Program
OSTI Identifier:
1302921
Grant/Contract Number:
AC05-00OR22725
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
SIAM Journal on Numerical Analysis
Additional Journal Information:
Journal Volume: 54; Journal Issue: 3; Journal ID: ISSN 0036-1429
Publisher:
Society for Industrial and Applied Mathematics
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Monte Carlo; multilevel; ltering; Kalman lter; ensemble Kalman lter

Citation Formats

Hoel, Hakon, Law, Kody J. H., and Tempone, Raul. Multilevel ensemble Kalman filtering. United States: N. p., 2016. Web. doi:10.1137/15M100955X.
Hoel, Hakon, Law, Kody J. H., & Tempone, Raul. Multilevel ensemble Kalman filtering. United States. doi:10.1137/15M100955X.
Hoel, Hakon, Law, Kody J. H., and Tempone, Raul. 2016. "Multilevel ensemble Kalman filtering". United States. doi:10.1137/15M100955X. https://www.osti.gov/servlets/purl/1302921.
@article{osti_1302921,
title = {Multilevel ensemble Kalman filtering},
author = {Hoel, Hakon and Law, Kody J. H. and Tempone, Raul},
abstractNote = {This study embeds a multilevel Monte Carlo sampling strategy into the Monte Carlo step of the ensemble Kalman filter (EnKF) in the setting of finite dimensional signal evolution and noisy discrete-time observations. The signal dynamics is assumed to be governed by a stochastic differential equation (SDE), and a hierarchy of time grids is introduced for multilevel numerical integration of that SDE. Finally, the resulting multilevel EnKF is proved to asymptotically outperform EnKF in terms of computational cost versus approximation accuracy. The theoretical results are illustrated numerically.},
doi = {10.1137/15M100955X},
journal = {SIAM Journal on Numerical Analysis},
number = 3,
volume = 54,
place = {United States},
year = 2016,
month = 6
}

Journal Article:
Free Publicly Available Full Text
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Citation Metrics:
Cited by: 2works
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  • The proof of convergence of the standard ensemble Kalman filter (EnKF) from Le Gland, Monbet, and Tran [Large sample asymptotics for the ensemble Kalman filter, in The Oxford Handbook of Nonlinear Filtering, Oxford University Press, Oxford, UK, 2011, pp. 598--631] is extended to non-Gaussian state-space models. In this paper, a density-based deterministic approximation of the mean-field limit EnKF (DMFEnKF) is proposed, consisting of a PDE solver and a quadrature rule. Given a certain minimal order of convergence κ between the two, this extends to the deterministic filter approximation, which is therefore asymptotically superior to standard EnKF for dimension d <more » 2κ. The fidelity of approximation of the true distribution is also established using an extension of the total variation metric to random measures. Lastly, this is limited by a Gaussian bias term arising from nonlinearity/non-Gaussianity of the model, which arises in both deterministic and standard EnKF. Numerical results support and extend the theory.« less
  • A new approach is presented for data assimilation using the ensemble adjustment Kalman filter (EAKF) technique for surface measurements of carbon monoxide in a single tracer version of the community air quality model. An implementation of the EAKF known as the Data Assimilation Research Testbed at the National Center for Atmospheric Research was used for developing the model. Three different sets of numerical experiments were performed to test the effectiveness of the procedure and the range of key parameters used in implementing the procedure. The model domain includes much of the northeastern United States. The first two numerical experiments usemore » idealized measurements derived from defined model runs, and the last test uses measurements of carbon monoxide from approximately 220 Air Quality System monitoring sites over the northeastern United States, maintained by the U.S. Environmental Protection Agency. In each case, the proposed method provided better results than the method without data assimilation.« less
  • Kalman filtering (KF) is used to postprocess numerical-model output to estimate systematic errors in surface ozone forecasts. It is implemented with a recursive algorithm that updates its estimate of future ozone-concentration bias by using past forecasts and observations. KF performance is tested for three types of ozone forecasts: deterministic, ensemble-averaged, and probabilistic forecasts. Eight photochemical models were run for 56 days during summer 2004 over northeastern USA and southern Canada as part of the International Consortium for Atmospheric Research on Transport and Transformation New England Air Quality (AQ) Study. The raw and KF-corrected predictions are compared with ozone measurements frommore » the Aerometric Information Retrieval Now data set, which includes roughly 360 surface stations. The completeness of the data set allowed a thorough sensitivity test of key KF parameters. It is found that the KF improves forecasts of ozone-concentration magnitude and the ability to predict rare events, both for deterministic and ensemble-averaged forecasts. It also improves the ability to predict the daily maximum ozone concentration, and reduces the time lag between the forecast and observed maxima. For this case study, KF considerably improves the predictive skill of probabilistic forecasts of ozone concentration greater than thresholds of 10 to 50 ppbv, but it degrades it for thresholds of 70 to 90 ppbv. Moreover, KF considerably reduces probabilistic forecast bias. The significance of KF postprocessing and ensemble-averaging is that they are both effective for real-time AQ forecasting. KF reduces systematic errors, whereas ensemble-averaging reduces random errors. When combined they produce the best overall forecast.« less
  • To reduce the computational load of the ensemble Kalman filter while maintaining its efficacy, an optimization algorithm based on the generalized eigenvalue decomposition method is proposed for identifying the most informative measurement subspace. When the number of measurements is large, the proposed algorithm can be used to make an effective tradeoff between computational complexity and estimation accuracy. This algorithm also can be extended to other Kalman filters for measurement subspace selection.