Time-Limited Balanced Truncation for Data Assimilation Problems

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 (Qian et al in Sci Comput 91:29, 2022). 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.

Automated summary

This paper discusses the application of balanced truncation to linear Gaussian Bayesian inference, particularly the 4D-Var method, and strengthens the connection between systems theory and data assimilation. The similarities between both types of data assimilation problems allow for the generalization of the state-of-the-art approach, proposing an enhanced method to balance Bayesian inference for unstable systems and improve numerical results for short observation periods.