State estimation in spatiotemporal chaos via low-rank StatFEM

Area of Science:

• Computational physics
• Applied mathematics
• Dynamical systems

Background:

• High-dimensional spatiotemporal chaos presents significant challenges for trajectory reconstruction.
• Model misspecification and computational expense hinder accurate state estimation in chaotic systems.

Purpose of the Study:

• To introduce a computationally feasible and robust framework for state estimation in high-dimensional chaotic systems.
• To address challenges posed by structural model errors and sparse, noisy observations.

Main Methods:

• Development of a Low-Rank Extended Rauch-Tung-Striebel Smoother (LR-ExRTSS) within the Statistical Finite Element (StatFEM) framework.
• Application of a grid-independent, retrospective scheme to the 2D anisotropic Kuramoto-Sivashinsky equation.
• Utilizing a combination of a square-root retrospective smoother and a forward-looking low-rank extended Kalman filter.

Main Results:

• The LR-ExRTSS successfully bridges the gap between misspecified physics and chaotic reality.
• The scheme effectively reintroduces transverse instabilities that were aggressively damped by the flawed predictive model.
• Spectral analysis confirmed the scheme's ability to span the true system's unstable-neutral subspace, irrespective of the model's prediction.

Conclusions:

• Low-rank Statistical Finite Element (StatFEM) provides a computationally feasible framework for robust state estimation.
• The LR-ExRTSS demonstrates efficacy in high-dimensional systems even under severe structural model errors.
• This approach offers a viable solution for tracking chaotic dynamics with limited and imperfect data.