Variational Bayesian method has been fully investigated in machine learning field. However, it is rarely used in inverse problems. Actually, I can only find the following references.
[1] B. Jin and J. Zou, Hierarchical Bayesian inference for ill-posed problems via variational method, Journal of Computational Physics, 229, 2010, 7317-7343.
[2] B. Jin, A variational Bayesian method to inverse problems with implusive noise, Journal of Computational Physics, 231, 2012, 423-435.
[3] Nilabja Guha, Xiaoqing Wu, Yalchin Efendiev, Bangti Jin, Bani K. Mallick, A Bayesian variational approach for inverse problems with skew-t error distributions, Journal of Computational Physics,301, 2015, 377-393.
[4] K. Yang, N. Guha, Y. Efendiev, and B. K. Malick, Bayesian and variational Bayesian approaches for flows in heterogeneous random media, Journal of Computational Physics, 345, 2017, 275-293.
[5] L. Gharsalli,H. Ayasso, B. Duchene, and A. Mohammad-Djafari, Inverse scattering in a Bayesian framework: application to microwave imaging for breast cancer detection, Inverse Problems, 30, 2014, 114011.
This repository will provide some implementations of the above papers, and I will provide the codes for my later research concerned with this topic.