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.

翻译：本文提出一种基于非中心参数化的无限维均值场变分推断（NCP-iMFVI）方法，用于求解分层贝叶斯反问题。该方法能够从近似后验分布中高效地生成可用估计值。为避免无限维分层方法中出现的互异奇异性问题，我们建立了非中心变分贝叶斯方法的严格理论。由于非中心参数化弱化了参数与超参数之间的关联，我们可以将超参数引入先验协方差算子特征分解的所有项中。同时揭示了NCP-iMFVI方法与中心参数化无限维分层方法之间的关系。将所提算法应用于由简单光滑方程、亥姆霍兹方程和稳态达西渗流方程控制的三类反问题。数值结果验证了理论发现，展示了求解由大规模线性和非线性统计反问题所构建的iMFVI问题的有效性，并验证了网格无关性。