A Structurally Informed Data Assimilation Approach for Nonlinear Partial Differential Equations

TL;DR

This paper introduces a structurally informed non-Gaussian prior for ensemble transform Kalman filtering that improves state estimation accuracy in nonlinear PDEs with discontinuities by exploiting gradient-based statistical information.

Contribution

The paper proposes a novel weighting matrix based on gradient second moments and clustering to incorporate discontinuity information into ETKF for nonlinear PDEs.

Findings

Improved accuracy over standard ETKF in shallow water equations.

Effective handling of discontinuities through gradient-based clustering.

Enhanced state estimates without additional inflation or localization.

Abstract

Ensemble transform Kalman filtering (ETKF) data assimilation is often used to combine available observations with numerical simulations to obtain statistically accurate and reliable state representations in dynamical systems. However, it is well known that the commonly used Gaussian distribution assumption introduces biases for state variables that admit discontinuous profiles, which are prevalent in nonlinear partial differential equations. This investigation designs a new structurally informed non-Gaussian prior that exploits statistical information from the simulated state variables. In particular, we construct a new weighting matrix based on the second moment of the gradient information of the state variable to replace the prior covariance matrix used for model/data compromise in the ETKF data assimilation framework. We further adapt our weighting matrix to include information in discontinuity regions via a clustering technique. Our numerical experiments demonstrate that this new approach yields more accurate estimates than those obtained using ETKF on shallow water equations, even when ETKF is enhanced with inflation and localization techniques.