Hierarchical sparse observation models and informative prior for Bayesian inference of spatially varying parameters

Abstract We develop a new computational approach for the speedy and accurate recovery of unknown spatially varying parameters of (stochastic) PDE models based on the sequentially arriving noisy observations of the evolving system variables. Two essential ingredients in our strategy for the identification of the target spatial fields are given by the Fourier diagonalization (FD) method and the hierarchical Bayesian model. We first apply the FD framework to perform an effectively parallelized computation in quantifying the propagating uncertainties of the dynamic variables in the Fourier space, and to facilitate a drastic saving of computational resources required for the Bayesian estimation of the associated parameters via sequential data assimilation. Yet our case study reveals that a very poor performance of the FD scheme occurs when the fast and slow variables coexist in the system evolution. The key observation is that, estimating the parameters in association with some slow variables, the FD-based Bayesian solver significantly underperforms compared to the remaining cases of faster variables. Due to this highly non-uniform discrepancy in the accuracy across distinct Fourier modes, the approximation of the parameter field cannot be so desired if the consequence is represented in the physical space. As one effort to circumvent this problem, we provide a systematic approach for a radical improvement of the naive FD technique. To do this, we make use of the hierarchical Bayesian model; it refers to the process of gradually enriching the knowledge of the unknown spatial fields from coarse to medium and to fine resolutions by using the Bayesian inference obtained at coarser levels to provide prior information for the estimation at finer levels. One major contribution of this paper is the development of a variant of the classical application of the multi-resolution approach through the close integration with the FD method, leading to the emergence of a new Bayesian paradigm for the data-driven parametric identification. Numerical experiments are performed to corroborate our demonstration concerning the efficacy and effectiveness of the proposed algorithm in obtaining a good degree of accuracy together with a significantly reduced computational cost.