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Optimization based methods for partially observed chaotic systems

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Title: Optimization based methods for partially observed chaotic systems
Authors: Paulin, D
Jasra, A
Crisan, DO
Beskos, A
Item Type: Journal Article
Abstract: In this paper we consider filtering and smoothing of partially observed chaotic dynamical systems that are discretely observed, with an additive Gaussian noise in the observation. These models are found in a wide variety of real applications and include the Lorenz 96’ model. In the context of a fixed observation interval T, observation time step h and Gaussian observation variance σ2Z, we show under assumptions that the filter and smoother are well approximated by a Gaussian with high probability when h and σ2Zh are sufficiently small. Based on this result we show that the maximum a posteriori (MAP) estimators are asymptotically optimal in mean square error as σ2Zh tends to 0. Given these results, we provide a batch algorithm for the smoother and filter, based on Newton’s method, to obtain the MAP. In particular, we show that if the initial point is close enough to the MAP, then Newton’s method converges to it at a fast rate. We also provide a method for computing such an initial point. These results contribute to the theoretical understanding of widely used 4D-Var data assimilation method. Our approach is illustrated numerically on the Lorenz 96’ model with state vector up to 1 million dimensions, with code running in the order of minutes. To our knowledge the results in this paper are the first of their type for this class of models.
Issue Date: 1-Jun-2019
Date of Acceptance: 9-Mar-2018
URI: http://hdl.handle.net/10044/1/58391
DOI: https://dx.doi.org/10.1007/s10208-018-9388-x
ISSN: 1615-3375
Publisher: Springer Verlag
Start Page: 485
End Page: 559
Journal / Book Title: Foundations of Computational Mathematics
Volume: 19
Issue: 3
Copyright Statement: © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Sponsor/Funder: Engineering and Physical Sciences Research Council
Funder's Grant Number: EP/N023781/1
Keywords: 01 Mathematical Sciences
08 Information and Computing Sciences
Numerical & Computational Mathematics
Publication Status: Published
Online Publication Date: 2018-04-25
Appears in Collections:Pure Mathematics
Mathematics
Faculty of Natural Sciences



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