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Ensemble Kalman methods for inverse problems

Iglesias, Marco A. and Law, Kody J. H. and Stuart, Andrew M. (2013) Ensemble Kalman methods for inverse problems. Inverse Problems, 29 (4). Art. No. 045001. ISSN 0266-5611. doi:10.1088/0266-5611/29/4/045001.

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The ensemble Kalman filter (EnKF) was introduced by Evensen in 1994 (Evensen 1994 J. Geophys. Res. 99 10143–62) as a novel method for data assimilation: state estimation for noisily observed time-dependent problems. Since that time it has had enormous impact in many application domains because of its robustness and ease of implementation, and numerical evidence of its accuracy. In this paper we propose the application of an iterative ensemble Kalman method for the solution of a wide class of inverse problems. In this context we show that the estimate of the unknown function that we obtain with the ensemble Kalman method lies in a subspace A spanned by the initial ensemble. Hence the resulting error may be bounded above by the error found from the best approximation in this subspace. We provide numerical experiments which compare the error incurred by the ensemble Kalman method for inverse problems with the error of the best approximation in A, and with variants on traditional least-squares approaches, restricted to the subspace A. In so doing we demonstrate that the ensemble Kalman method for inverse problems provides a derivative-free optimization method with comparable accuracy to that achieved by traditional least-squares approaches. Furthermore, we also demonstrate that the accuracy is of the same order of magnitude as that achieved by the best approximation. Three examples are used to demonstrate these assertions: inversion of a compact linear operator; inversion of piezometric head to determine hydraulic conductivity in a Darcy model of groundwater flow; and inversion of Eulerian velocity measurements at positive times to determine the initial condition in an incompressible fluid.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper
Iglesias, Marco A.0000-0002-8952-717X
Additional Information:© 2013 IOP. Received 13 September 2012, in final form 15 February 2013. Published 13 March 2013. This was supported by the ERC, EPSRC and ONR.
Funding AgencyGrant Number
European Research Council (ERC)UNSPECIFIED
Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
Office of Naval Research (ONR)UNSPECIFIED
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Andrew StuartJ99
Issue or Number:4
Record Number:CaltechAUTHORS:20160727-180147046
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Official Citation:Marco A Iglesias et al 2013 Inverse Problems 29 045001
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:69265
Deposited By: Linda Taddeo
Deposited On:28 Jul 2016 23:45
Last Modified:12 Jul 2022 19:47

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