Continuous time limit of the stochastic ensemble Kalman inversion: strong convergence analysis

  • The ensemble Kalman inversion (EKI) method is a method for the estimation of unknown parameters in the context of (Bayesian) inverse problems. The method approximates the underlying measure by an ensemble of particles and iteratively applies the ensemble Kalman update to evolve (the approximation of the) prior into the posterior measure. For the convergence analysis of the EKI it is common practice to derive a continuous version, replacing the iteration with a stochastic differential equation. In this paper we validate this approach by showing that the stochastic EKI iteration converges to paths of the continuous time stochastic differential equation by considering both the nonlinear and linear setting, and we prove convergence in probability for the former and convergence in moments for the latter. The methods employed do not rely on the specific structure of the ensemble Kalman method and can also be applied to the analysis of more general numerical schemes for stochasticThe ensemble Kalman inversion (EKI) method is a method for the estimation of unknown parameters in the context of (Bayesian) inverse problems. The method approximates the underlying measure by an ensemble of particles and iteratively applies the ensemble Kalman update to evolve (the approximation of the) prior into the posterior measure. For the convergence analysis of the EKI it is common practice to derive a continuous version, replacing the iteration with a stochastic differential equation. In this paper we validate this approach by showing that the stochastic EKI iteration converges to paths of the continuous time stochastic differential equation by considering both the nonlinear and linear setting, and we prove convergence in probability for the former and convergence in moments for the latter. The methods employed do not rely on the specific structure of the ensemble Kalman method and can also be applied to the analysis of more general numerical schemes for stochastic differential equations.show moreshow less

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Metadaten
Author:Dirk BlömkerORCiDGND, Claudia Schillings, Philipp Wacker, Simon Weissmann
URN:urn:nbn:de:bvb:384-opus4-1044121
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/104412
Parent Title (English):SIAM Journal on Numerical Analysis
Publisher:Society for Industrial & Applied Mathematics (SIAM)
Place of publication:Philadelphia, PA
Type:Article
Language:English
Year of first Publication:2022
Publishing Institution:Universität Augsburg
Release Date:2023/05/16
Volume:60
Issue:6
First Page:3181
Last Page:3215
DOI:https://doi.org/10.1137/21M1437561
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Nichtlineare Analysis
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):Deutsches Urheberrecht