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

TL;DR

This paper introduces a non-centered variational Bayesian method for hierarchical inverse problems in PDEs, improving efficiency and theoretical robustness in infinite-dimensional Bayesian inference.

Contribution

It develops a rigorous non-centered variational Bayesian framework for infinite-dimensional inverse problems, addressing mutual singularity issues and establishing relationships with centered approaches.

Findings

Efficiently solves large-scale linear and nonlinear inverse problems.

Demonstrates mesh-independent property in numerical experiments.

Validates theoretical advantages through applications to PDE-based inverse problems.

Abstract

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.