Statistical finite elements for misspecified models.

Area of Science:

• Computational science and engineering
• Applied mathematics
• Physics

Background:

• Nonlinear, time-dependent phenomena pose significant modeling challenges.
• Traditional methods may struggle with data assimilation and uncertainty quantification.
• Finite element methods (FEM) are widely used for complex physical systems.

Purpose of the Study:

• To introduce a novel statistical finite element method (sFEM) for nonlinear, time-dependent problems.
• To integrate Bayesian inference and FEM within a state-space modeling framework.
• To demonstrate the method's applicability and effectiveness using real-world data.

Main Methods:

• Developed a Bayesian approach leveraging FEM to formulate a nonlinear Gaussian state-space model.
• Implemented the sFEM using extended and ensemble Kalman filter algorithms.
• Validated the method with simulations and experimental data for nonlinear internal waves (solitons).

Main Results:

• The sFEM effectively updates solutions in a filtering framework when data is available.
• Demonstrated successful application to the Korteweg-de Vries equation (solitons).
• Showcased the method's generality with Burgers and Kuramoto-Sivashinsky equations.

Conclusions:

• The presented statistical finite element method offers a robust framework for analyzing nonlinear, time-dependent phenomena.
• The Bayesian filtering approach enhances predictive accuracy and uncertainty quantification.
• This versatile method is applicable across diverse scientific and engineering disciplines requiring FEM.