Time-limited Balanced Truncation for Data Assimilation Problems

TL;DR

This paper explores the use of time-limited balanced truncation in data assimilation, specifically in linear Gaussian Bayesian inference and 4D-Var, to improve model reduction and handle unstable systems more effectively.

Contribution

It introduces a novel application of time-limited balanced truncation to Bayesian inference and 4D-Var, enhancing stability handling and numerical performance for short observation periods.

Findings

Enables balancing Bayesian inference for unstable systems.

Improves numerical results for short observation periods.

Generalizes the use of arbitrary prior covariances as reachability Gramians.

Abstract

Balanced truncation is a well-established model order reduction method which has been applied to a variety of problems. Recently, a connection between linear Gaussian Bayesian inference problems and the system-theoretic concept of balanced truncation has been drawn. Although this connection is new, the application of balanced truncation to data assimilation is not a novel idea: it has already been used in four-dimensional variational data assimilation (4D-Var). This paper discusses the application of balanced truncation to linear Gaussian Bayesian inference, and, in particular, the 4D-Var method, thereby strengthening the link between systems theory and data assimilation further. Similarities between both types of data assimilation problems enable a generalisation of the state-of-the-art approach to the use of arbitrary prior covariances as reachability Gramians. Furthermore, we propose an enhanced approach using time-limited balanced truncation that allows to balance Bayesian inference for unstable systems and in addition improves the numerical results for short observation periods.