Journal
SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION
Volume 3, Issue 1, Pages 823-851Publisher
SIAM PUBLICATIONS
DOI: 10.1137/140981319
Keywords
inverse problems; Bayesian inverse problem; Bayes estimator; Kalman filter; ensemble Kalman filter; polynomial chaos; conditional distribution
Funding
- DFG priority programme 1324
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We analyze the ensemble and polynomial chaos Kalman filters applied to nonlinear stationary Bayesian inverse problems. In a sequential data assimilation setting, such stationary problems arise in each step of either filter. We give a new interpretation of the approximations produced by these two popular filters in the Bayesian context and prove that, in the limit of large ensemble or high polynomial degree, both methods yield approximations which converge to a well-defined random variable termed the analysis random variable. We then show that this analysis variable is more closely related to a specific linear Bayes estimator than to the solution of the associated Bayesian inverse problem given by the posterior measure. This suggests limited or at least guarded use of these generalized Kalman filter methods for the purpose of uncertainty quantification.
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