A Bayesian-variational cyclic method for solving estimation problems characterized by non-uniqueness (equifinality)

In this paper a new method for solving complex estimation problems for systems governed by partial differential equations and characterized by non-uniqueness (or equifinality) is presented. For such problems, the usefulness of the MAP estimates is doubtful, especially if the available prior information is very limited or not plausible in general. Thus, the posterior mean or median estimates would be more appropriate, which implies that the Bayesian approach has to be preferred. Since the direct use of the Bayesian approach involving computationally expensive models is not feasible a hybrid method which combines the Bayesian and variational elements in a unique way is suggested. The main idea of the method is to find one particular posterior mode out of a possibly infinite set, which would have the global properties (its first moments, for example) consistent with the available prior information and close to those of the posterior mean. The method is implemented in the form of a cyclic algorithm including the Bayesian estimation part for the lumped global (hidden, latent) variables describing the moments of the spatially distributed or time-dependent variables and the variational estimation part for the invariant 'shape' functions. The method has been numerically validated for two different applications: one including the Saint-Venant hydraulic model, and another one including the nonlinear convection-diffusion transport model. The results for the former are reported in a separate paper (reference provided), for the latter - in the present paper. These results confirm the expected properties of the suggested algorithm, such as improved robustness and accuracy, in comparison to the MAP estimator represented by the variational estimation algorithm.