Statistical finite elements for misspecified models

We present a statistical finite element method for nonlinear, time-dependent phenomena, illustrated in the context of nonlinear internal waves (solitons). We take a Bayesian approach and leverage the finite element method to cast the statistical problem as a nonlinear Gaussian state–space model, upd...

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Bibliographic Details
Published in:Proceedings of the National Academy of Sciences - PNAS 2021-01, Vol.118 (2), p.1-6
Main Authors: Duffin, Connor, Cripps, Edward, Stemler, Thomas, Girolami, Mark
Format: Article
Language:English
Subjects:
Algorithms
Bayesian analysis
Burgers equation
Differential equations
Finite element analysis
Finite element method
Internal waves
Kalman filters
Mathematical models
Nonlinear equations
Partial differential equations
Physical Sciences
Solitary waves
Statistical analysis
Statistics
Time dependence
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Summary:We present a statistical finite element method for nonlinear, time-dependent phenomena, illustrated in the context of nonlinear internal waves (solitons). We take a Bayesian approach and leverage the finite element method to cast the statistical problem as a nonlinear Gaussian state–space model, updating the solution, in receipt of data, in a filtering framework. The method is applicable to problems across science and engineering for which finite element methods are appropriate. The Korteweg–de Vries equation for solitons is presented because it reflects the necessary complexity while being suitably familiar and succinct for pedagogical purposes. We present two algorithms to implement this method, based on the extended and ensemble Kalman filters, and demonstrate effectiveness with a simulation study and a case study with experimental data. The generality of our approach is demonstrated in SI Appendix, where we present examples from additional nonlinear, time-dependent partial differential equations (Burgers equation, Kuramoto–Sivashinsky equation).
ISSN:0027-8424
1091-6490
DOI:10.1073/pnas.2015006118