Convergence and Application of a Modified Iteratively Regularized Gauss-Newton Algorithm

We establish theoretical convergence results for an Iteratively Regularized Gauss Newton (IRGN) algorithm with a specific Tikhonov regularization. This Tikhnov regularization, which uses a seminorm generated by a linear operator, is motivated by mapping of the minimization variables to physical space which exposes the different scales of the parameters and therefore also suggests appropriate weighting of the regularization terms with respect to the parameter spaces. The basic convergence result uses an a posteriori stopping rule and a modified source condition, without any restriction on the nonlinearity of the operator. We illustrate our theoretical results using simulations for a one dimensional version of the exponentially ill-posed optical tomography inverse problem for which the parameter space depends on diffusion coefficient D and absorption coefficient μ which are on very different scales. We conclude that the new method contributes greater flexibility for implementations of IRGN solutions of ill-posed inverse problems in which differing scales in physical space hinder standard IRGN inversions.