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This paper provides a detailed theoretical analysis of methods to approximatethe solutions of high-dimensional (>10^6) linear Bayesian problems. An optimallow-rank projection that maximizes the information content of the Bayesianinversion is proposed and efficiently constructed using a scalable randomizedSVD algorithm. Useful optimality results are established for the associatedposterior error covariance matrix and posterior mean approximations, which arefurther investigated in a numerical experiment consisting of a large-scaleatmospheric tracer transport source-inversion problem. This method proves to bea robust and efficient approach to dimension reduction, as well as a naturalframework to analyze the information content of the inversion. Possibleextensions of this approach to the non-linear framework in the context ofoperational numerical weather forecast data assimilation systems based on theincremental 4D-Var technique are also discussed, and a detailed implementationof a new Randomized Incremental Optimal Technique (RIOT) for 4D-Var algorithmsleveraging our theoretical results is proposed.

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