# Variational Bayes' method for functions with applications to some inverse problems

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 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 sampling type method. 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. General infinite-dimensional mean-field approximate theory has been established, and has been applied to abstract linear inverse problems with Gaussian and Laplace noise assumptions. Finally, some numerical examples are given which illustrate the effectiveness of the proposed approach.

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