  This paper proposes a non-centered parameterization based
infinite-dimensional mean-field variational inference (NCP-iMFVI) approach 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
infinite-dimensional hierarchical approach, we propose a rigorous theory of the
non-centered 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 nonlinear statistical inverse
problems, and verify the mesh-independent property.
