2012/09/30 by Marco Iglesias, Marco A. Iglesias, Kody J. H. Law +1 · 8 citations
Earth and Planetary Sciences · Engineering · Mathematics · #Algorithm #Applied mathematics #Data assimilation #Ensemble Kalman filter #Extended Kalman filter #Inverse #Inverse problem #Kalman filter #Mathematical analysis #Mathematical optimization #Mathematics #Meteorological Phenomena and Simulations #Reservoir Engineering and Simulation Methods #Seismic Imaging and Inversion Techniques #Statistics #Subspace topology #math.OC
paper · pdf · doi:10.1088/0266-5611/29/4/045001
published as Inverse Problems 29 (2013) 045001
arxiv created 2013/03/01 · openalex publication_date 2013/03/13 · arxiv updated 2013/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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 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 , and with variants on traditional least-squares approaches, restricted to the subspace . 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.