  Bayesian approach, as a useful tool for quantifying uncertainties, has been
widely used for solving inverse problems of partial differential equations
(PDEs). One of the key difficulties for employing Bayesian approach for the
issue is how to extract information from the posterior probability measure.
Variational Bayes' method (VBM) is firstly and broadly studied in the field of
machine learning, which has the ability to extract posterior information
approximately by using much lower computational resources compared with the
conventional sampling type methods. In this paper, we generalize the usual
finite-dimensional VBM to infinite-dimensional space, which makes the usage of
VBM for inverse problems of PDEs rigorously. We further establish general
infinite-dimensional mean-field approximate theory, and apply this theory to
abstract linear inverse problems with Gaussian and Laplace noise assumptions.
The results on some numerical examples substantiate the effectiveness of the
proposed approach.
