Ensemble Kalman methods for inverse problems

By M. Iglesias, Kody J. H. Law, Andrew M. Stuart. 2012

460 citations recorded by the source corpus.

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.

Subfields: Atmospheric Science, Geophysics, Ocean Engineering

View Paper: https://www.semanticscholar.org/paper/dbdf03e84525d054c129c85529c0f198b0e49078

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Citations in the corpus

Earth System Modeling 2.0: A Blueprint for Models That Learn From Observations and Targeted High‐Resolution Simulations (2017, methodology): “…such uncertainty information is qualitative at best: the Kalman methods invoke Gaussian assumptions, which may not be justified, and even if the Gaussian approximation holds, the ensemble sizes needed for uncertainty quantification may not be practical [Law and Stuart, 2012; Iglesias et al., 2013].”
Turbulence Modeling in the Age of Data (2018, methodology): “All these approaches involve large-scale statistical inference using adjoint-based algorithms (Giles et al. 2003) or derivative-free, iterative Ensemble Kalman method (Iglesias et al. 2013).”
Quantification of model uncertainty in RANS simulations: A review (2018, methodology): “Alternatively, the optimization problem in Equation (31) can also be solved by the iterative ensemble Kalman method [85], which can be considered a derivative-free optimization that uses the state covariance, estimated from Monte Carlo samples, instead of the Jacobian.”