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Iterated Kalman methodology for inverse problems

Huang, Daniel Zhengyu and Schneider, Tapio and Stuart, Andrew M. (2022) Iterated Kalman methodology for inverse problems. Journal of Computational Physics, 463 . Art. No. 111262. ISSN 0021-9991. doi:10.1016/

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This paper is focused on the optimization approach to the solution of inverse problems. We introduce a stochastic dynamical system in which the parameter-to-data map is embedded, with the goal of employing techniques from nonlinear Kalman filtering to estimate the parameter given the data. The extended Kalman filter (which we refer to as ExKI in the context of inverse problems) can be effective for some inverse problems approached this way, but is impractical when the forward map is not readily differentiable and is given as a black box, and also for high dimensional parameter spaces because of the need to propagate large covariance matrices. Application of ensemble Kalman filters, for example use of the ensemble Kalman inversion (EKI) algorithm, has emerged as a useful tool which overcomes both of these issues: it is derivative free and works with a low-rank covariance approximation formed from the ensemble. In this paper, we work with the ExKI, EKI, and a variant on EKI which we term unscented Kalman inversion (UKI). The paper contains two main contributions. Firstly, we identify a novel stochastic dynamical system in which the parameter-to-data map is embedded. We present theory in the linear case to show exponential convergence of the mean of the filtering distribution to the solution of a regularized least squares problem. This is in contrast to previous work in which the EKI has been employed where the dynamical system used leads to algebraic convergence to an unregularized problem. Secondly, we show that the application of the UKI to this novel stochastic dynamical system yields improved inversion results, in comparison with the application of EKI to the same novel stochastic dynamical system. The numerical experiments include proof-of-concept linear examples and various applied nonlinear inverse problems: learning of permeability parameters in subsurface flow; learning the damage field from structure deformation; learning the Navier-Stokes initial condition from solution data at positive times; learning subgrid-scale parameters in a general circulation model (GCM) from time-averaged statistics.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper
Schneider, Tapio0000-0001-5687-2287
Stuart, Andrew M.0000-0001-9091-7266
Alternate Title:Unscented Kalman Inversion
Additional Information:© 2022 Elsevier Inc. Received 1 February 2021, Revised 26 April 2022, Accepted 27 April 2022, Available online 4 May 2022, Version of Record 10 May 2022. This work was supported by the generosity of Eric and Wendy Schmidt by recommendation of the Schmidt Futures program and by the National Science Foundation (NSF, award AGS-1835860). A.M.S. was also supported by the Office of Naval Research (award N00014-17-1-2079). The authors thank Sebastian Reich and anonymous reviewers for helpful comments on an earlier draft. CRediT authorship contribution statement: Daniel Zhengyu Huang: Methodology, Software, Visualization, Writing – original draft, Writing – review & editing. Tapio Schneider: Conceptualization, Funding acquisition, Writing – original draft. Andrew M. Stuart: Conceptualization, Funding acquisition, Writing – original draft, Writing – review & editing. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Funding AgencyGrant Number
Schmidt Futures ProgramUNSPECIFIED
Office of Naval Research (ONR)N00014-17-1-2079
Subject Keywords:Inverse problem; Derivative-free optimization; Extended Kalman methods; Ensemble Kalman methods; Unscented Kalman methods; Interacting particle systems
Record Number:CaltechAUTHORS:20210719-210149563
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Official Citation:Daniel Zhengyu Huang, Tapio Schneider, Andrew M. Stuart, Iterated Kalman methodology for inverse problems, Journal of Computational Physics, Volume 463, 2022, 111262, ISSN 0021-9991,
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:109922
Deposited By: George Porter
Deposited On:19 Jul 2021 22:53
Last Modified:10 May 2022 22:09

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