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This paper proposes a non-centered parameterization basedinfinite-dimensional mean-field variational inference (NCP-iMFVI) approach forsolving the hierarchical Bayesian inverse problems. This method can generateavailable estimates from the approximated posterior distribution efficiently.To avoid the mutually singular obstacle that occurred in theinfinite-dimensional hierarchical approach, we propose a rigorous theory of thenon-centered variational Bayesian approach. Since the non-centeredparameterization weakens the connection between the parameter and thehyper-parameter, we can introduce the hyper-parameter to all terms of theeigendecomposition of the prior covariance operator. We also show therelationships between the NCP-iMFVI and infinite-dimensional hierarchicalapproaches with centered parameterization. The proposed algorithm is applied tothree inverse problems governed by the simple smooth equation, the Helmholtzequation, and the steady-state Darcy flow equation. Numerical results confirmour theoretical findings, illustrate the efficiency of solving the iMFVIproblem formulated by large-scale linear and nonlinear statistical inverseproblems, and verify the mesh-independent property.

Non-centered parametric variational Bayes' approach for hierarchical inverse problems of partial differential equations