NON-CENTERED PARAMETRIC VARIATIONAL BAYES' APPROACH FOR HIERARCHICAL INVERSE PROBLEMS OF PARTIAL DIFFERENTIAL EQUATIONS

. This paper proposes a non-centered parameterization based for solving the hierarchical Bayesian inverse problems. This method can generate available estimates from the approximated posterior distribution efficiently. To avoid the mutually singular obstacle that occurred in the infinitedimensional hierarchical approach, we propose a rigorous theory of the noncentered variational Bayesian approach. Since the non-centered parameterization weakens the connection between the parameter and the hyper-parameter, we can introduce the hyper-parameter to all terms of the eigendecomposition of the prior covariance operator. We also show the relationships between the NCP-iMFVI and infinite-dimensional hierarchical approaches with centered parameterization. The proposed algorithm is applied to three inverse problems governed by the simple smooth equation, the Helmholtz equation, and the steady-state Darcy flow equation. Numerical results confirm our theoretical findings, illustrate the efficiency of solving the iMFVI problem formulated by large-scale linear and non-linear statistical inverse problems, and verify the mesh-independent property.