Optimal and scalable methods to approximate the solutions of large-scale Bayesian problems: Theory and application to atmospheric inversions and data assimilation

TL;DR

This paper introduces a scalable, optimal low-rank approximation method for high-dimensional Bayesian problems, with applications to atmospheric data assimilation, improving efficiency and information content analysis.

Contribution

It proposes a novel randomized SVD-based approach for optimal dimension reduction in large-scale Bayesian problems, including a new RIOT method for 4D-Var data assimilation.

Findings

Effective in large-scale atmospheric inversion

Provides theoretical bounds for posterior error covariance

Demonstrates robustness and efficiency in numerical experiments

Abstract

This paper provides a detailed theoretical analysis of methods to approximate the solutions of high-dimensional (>10^6) linear Bayesian problems. An optimal low-rank projection that maximizes the information content of the Bayesian inversion is proposed and efficiently constructed using a scalable randomized SVD algorithm. Useful optimality results are established for the associated posterior error covariance matrix and posterior mean approximations, which are further investigated in a numerical experiment consisting of a large-scale atmospheric tracer transport source-inversion problem. This method proves to be a robust and efficient approach to dimension reduction, as well as a natural framework to analyze the information content of the inversion. Possible extensions of this approach to the non-linear framework in the context of operational numerical weather forecast data assimilation systems based on the incremental 4D-Var technique are also discussed, and a detailed implementation of a new Randomized Incremental Optimal Technique (RIOT) for 4D-Var algorithms leveraging our theoretical results is proposed.