Multilevel ensemble Kalman filtering for spatio-temporal processes
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Numerische Mathematik
Multilevel ensemble Kalman filtering for spatio-temporal processes Alexey Chernov1 · Håkon Hoel2 · Kody J. H. Law3 · Fabio Nobile4 · Raul Tempone2,5 Received: 21 February 2018 / Revised: 19 October 2020 / Accepted: 22 October 2020 © The Author(s) 2020
Abstract We design and analyse the performance of a multilevel ensemble Kalman filter method (MLEnKF) for filtering settings where the underlying state-space model is an infinitedimensional spatio-temporal process. We consider underlying models that needs to be simulated by numerical methods, with discretization in both space and time. The multilevel Monte Carlo sampling strategy, achieving variance reduction through pairwise coupling of ensemble particles on neighboring resolutions, is used in the samplemoment step of MLEnKF to produce an efficent hierarchical filtering method for spatio-temporal models. Under sufficent regularity, MLEnKF is proven to be more efficient for weak approximations than EnKF, asymptotically in the large-ensemble and fine-numerical-resolution limit. Numerical examples support our theoretical findings.
Inquires about this work should be directed to Håkon Hoel ([email protected]).
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Kody J. H. Law [email protected] Alexey Chernov [email protected] Håkon Hoel [email protected] Fabio Nobile [email protected] Raul Tempone [email protected]
1
Institute for Mathematics, Carl von Ossietzky University Oldenburg, Oldenburg, Germany
2
Chair of Mathematics for Uncertainty Quantification, RWTH Aachen University, Aachen, Germany
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Department of Mathematics, University of Manchester, Manchester, UK
4
Institute of Mathematics, École polytechnique fédérale de Lausanne, Lausanne, Switzerland
5
Applied Mathematics and Computational Sciences, KAUST, Thuwal, Saudi Arabia
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A. Chernov et al.
Mathematics Subject Classification 65C30 · 65Y20
1 Introduction Filtering refers to the sequential estimation of the state u and/or parameters of a system through sequential incorporation of online data y. The most complete estimation of the state u n at time n is given by its probability distribution conditional on the observations up to the given time P(du n |y1 , . . . , yn ) [2,27]. For linear Gaussian systems, the analytical solution may be given in closed form via update formulae for the mean and covariance known as the Kalman filter [31]. More generally, however, closed form solutions typically are not known. One must therefore resort to either algorithms which approximate the probabilistic solution by leveraging ideas from control theory in the data assimilation community [27,32], or Monte Carlo methods to approximate the filtering distribution itself [2,11,15]. The ensemble Kalman filter (EnKF) [9,17,35] combines elements of both approaches. In the linear Gaussian case it converges to the Kalman filter solution in the large-ensemble limit [41], and even in the nonlinear case, under suitable assumptions it converges [36,37] to a limit which is optimal among those which incorporate
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