论文标题
非中心的参数变分贝叶斯的方法用于部分微分方程的分层反问题
Non-centered parametric variational Bayes' approach for hierarchical inverse problems of partial differential equations
论文作者
论文摘要
本文提出了一种基于非中心参数化的无限维平均均值变异推理（NCP-IMFVI）方法，用于解决层次结构的贝叶斯逆问题。该方法可以有效地从近似后验分布中产生可用的估计值。为了避免在无限尺寸分层方法中发生的相互奇异障碍，我们提出了一种严格的对非中心变分贝叶斯方法的理论。由于非中心的参数化削弱了参数和高参数之间的连接，因此我们可以将超参数引入先前协方差运算符的特征分类的所有术语。我们还显示了NCP-IMFVI与无限维层次结构方法之间的关系。所提出的算法应用于由简单的平滑方程，Helmholtz方程和稳态DARCY流程方程控制的三个反问题。数值结果证实了我们的理论发现，说明了解决由大规模线性和非线性统计反问题提出的IMFVI问题的效率，并验证了与网格无关的属性。
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.
