The Study Of Bayesian-Sparsity Constraint Algorithm For Ill-Posed Problems

Since the regularization method has been widely used in geophysics, image reconstruction, signal processing, biomedical engineering, control theory and many other fields, the study of ill-posed problems algorithms has been paid attention by more and more people, and greatly promoted the development of the theory and practice of solving an ill-posed problem.This paper investigates the ill-posed problem of Bayesian-sparsity constraint algorithm. The failure to depend on the data continuously lead to a small perturbation in the data may cause an enormous deviation of the solution. Recently, the sparsity-constraint in the Bayesian framework for inverse problems has attracted many authors’interests. This method can be applied to image restoration. When introducing sparse constraint under Bayesian framework, an urgent problem still needs to be solved, a practical obstacle for these is the lack of fast posterior sampling algorithms for sparse Bayesian inversion.This paper is based on Gibbs algorithm of Markov chain Monte Carlo(MCMC) sampling algorithms, we design a single component Gibbs MCMC sampler for priors relying on L1-norms,and the specific process of the algorithm is given.To verify the validity of the method, this paper applys the design method to two types of linear and nonlinear inverse problems. The numerical results showed that the Bayesian sparsity constraint algorithm can be used for solving ill-posed problem effectively, and it will provide theoretical support for the various types of inverse problems.