Robust algorithms for filtering and parameter optimization in inverse problems

The focus of this paper is ill-posed inverse problems. We emphasized on the Tikhonov´s functional form of regularization from both numerical and Statistical methods viewpoints. We further extended the concept of regularization to the Bayesian methods framework. The Bayesian paradigm provided a general unified framework to the ill-posed inverse problem. Due to the computational complexity in estimating the parameters of the Bayesian model, the variational methods approach was used as an alternative. The problem of parameter estimation in the Bayesian framework was reduced to an optimization problem through the variational methods approach. In effect, we showed that the optimum of the Bayesian unified approach for the ill-posed inverse problem corresponds to the optimum of the variational methods approach.