Iglesias, Marco A. and Lin, Kui and Lu, Shuai and Stuart, Andrew M. (2017) Filter Based Methods For Statistical Linear Inverse Problems. Communications in Mathematical Sciences , 15 (7). pp. 1867-1896. ISSN 1539-6746. https://resolver.caltech.edu/CaltechAUTHORS:20161221-113147238
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Abstract
Ill-posed inverse problems are ubiquitous in applications. Understanding of algorithms for their solution has been greatly enhanced by a deep understanding of the linear inverse problem. In the applied communities ensemble-based filtering methods have recently been used to solve inverse problems by introducing an artificial dynamical system. This opens up the possibility of using a range of other filtering methods, such as 3DVAR and Kalman based methods, to solve inverse problems, again by introducing an artificial dynamical system. The aim of this paper is to analyze such methods in the context of the linear inverse problem. Statistical linear inverse problems are studied in the sense that the observational noise is assumed to be derived via realization of a Gaussian random variable. We investigate the asymptotic behavior of filter based methods for these inverse problems. Rigorous convergence rates are established for 3DVAR and for the Kalman filters, including minimax rates in some instances. Blowup of 3DVAR and a variant of its basic form is also presented, and optimality of the Kalman filter is discussed. These analyses reveal a close connection between (iterated) regularization schemes in deterministic inverse problems and filter based methods in data assimilation. Numerical experiments are presented to illustrate the theory.
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Additional Information: | © 2017 International Press. Paper received on 12 June 2016; Paper accepted on 6 May 2017. Andrew M. Stuart is funded by EPSRC (under the Programme Grant EQUIP), DARPA (under EQUiPS) and ONR. Shuai Lu is supported by Special Funds for Major State Basic Research Projects of China (2015CB856003), NSFC (91130004, 11522108), Shanghai Science and Technology Commission Grant (14QA1400400) and the Programme of Introducing Talents of Discipline to Universities (number B08018), China. | ||||||||||||||||||
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Subject Keywords: | Kalman filter, 3DVAR, statistical inverse problems, artificial dynamics | ||||||||||||||||||
Issue or Number: | 7 | ||||||||||||||||||
Classification Code: | 2010 Mathematics Subject Classification: 47A52, 65J22, 93E11 | ||||||||||||||||||
Record Number: | CaltechAUTHORS:20161221-113147238 | ||||||||||||||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20161221-113147238 | ||||||||||||||||||
Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||||||||||
ID Code: | 73078 | ||||||||||||||||||
Collection: | CaltechAUTHORS | ||||||||||||||||||
Deposited By: | Linda Taddeo | ||||||||||||||||||
Deposited On: | 21 Dec 2016 19:49 | ||||||||||||||||||
Last Modified: | 03 Oct 2019 16:24 |
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