Reduced Bayesian Inversion

Bayesian inversion of models with large state and parameter spaces proves to be computationally complex. A combined state and parameter reduction can significantly decrease the computational time and cost required for the parameter estimation. The presented technique is based on the well-known balanced truncation approach. Classically, the balancing of the controllability and observability gramians allows a truncation of discardable states. Here the underlying model, being a linear or nonlinear control system, is augmented by constant parameter states and parameter inputs, which allows the concurrent computation of parameter identifiability. The resulting joint gramian (the cross gramian of an augmented system), or the identifiability gramian (the observability gramian of an augmented input-free system) are partitioned into a state- and a parameter-related sub-gramian. After a balancing procedure of the state-sub-gramian and the proper orthogonal decomposition of the provided experimental data, states and parameters are truncated based on singular values of the associated gramians. This approach is inspired by the goal-oriented method from, where balanced truncation applied is applied in a similar manner, balancing experimental observability and prediction observability. Utilizing the empirical gramian framework the method is demonstrated with numerical experiments.