Optimal and scalable methods to approximate the solutions of large‐scale Bayesian problems: theory and application to atmospheric inversion and data assimilation

Abstract

This article provides a detailed theoretical analysis of methods to approximate the solutions of high‐dimensional (>106) 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. Optimality results for the associated posterior error covariance matrix and posterior mean approximations obtained in previous studies are revisited and tested 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 nonlinear 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.