Maximum entropy regularization for Fredholm integral equations of the first kind
SIAM Journal on Mathematical Analysis
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
An Introduction to Variational Methods for Graphical Models
Machine Learning
Inverse Problem Theory and Methods for Model Parameter Estimation
Inverse Problem Theory and Methods for Model Parameter Estimation
Statistics and Computing
Monte Carlo Strategies in Scientific Computing
Monte Carlo Strategies in Scientific Computing
Variational Bayesian inference for a nonlinear forward model
IEEE Transactions on Signal Processing
Parameter Estimation in TV Image Restoration Using Variational Distribution Approximation
IEEE Transactions on Image Processing
A variational Bayesian method to inverse problems with impulsive noise
Journal of Computational Physics
Solving inverse problems by decomposition, classification and simple modeling
Information Sciences: an International Journal
| Hi-index | 31.45 |
This paper investigates a novel approximate Bayesian inference procedure for numerically solving inverse problems. A hierarchical formulation which determines automatically the regularization parameter and the noise level together with the inverse solution is adopted. The framework is of variational type, and it can deliver the inverse solution and regularization parameter together with their uncertainties calibrated. It approximates the posteriori probability distribution by separable distributions based on Kullback-Leibler divergence. Two approximations are derived within the framework, and some theoretical properties, e.g. variance estimate and consistency, are also provided. Algorithms for their efficient numerical realization are described, and their convergence properties are also discussed. Extensions to nonquadratic regularization/nonlinear forward models are also briefly studied. Numerical results for linear and nonlinear Cauchy-type problems arising in heat conduction with both smooth and nonsmooth solutions are presented for the proposed method, and compared with that by Markov chain Monte Carlo. The results illustrate that the variational method can faithfully capture the posteriori distribution in a computationally efficient way.