Joint additive Kullback-Leibler residual minimization and regularization for linear inverse problems

For the approximate solution of ill-posed inverse problems, the formulation of a regularization functional involves two separate decisions: the choice of the residual minimizer and the choice of the regularizor. In this paper, the Kullback-Leibler functional is used for both. The resulting regularization method can solve problems for which the operator and the observational data are positive along with the solution, as occur in many inverse problem applications. Here, existence, uniqueness, convergence and stability for the regularization approximations are established under quite natural regularity conditions. Convergence rates are obtained by using an a priori strategy.