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The maximum a posteriori (MAP) estimates for linear inverse problems are studied using hierarchical Gaussian models. The stability of this point estimate is considered with respect to different discretizations. We analyze the phenomena which appear when the discretization becomes finer. An edge-preserving Bayesian reconstruction method for signal restoration problems is introduced and studied with arbitrarily fine discretization. Moreover, different noise asymptotics are considered for the inverse problem. We show that the maximum a posteriori and conditional mean estimates converge under different conditions. Finally, we discuss connection of this method to Mumford–Shah functional. This paper reviews results from [6]. [ABSTRACT FROM AUTHOR]

The preface gives a summary of the session on Statistical methods in inverse problems held at the 8th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2010). The papers presented in this session are reviewed in this preface. [ABSTRACT FROM AUTHOR]