Convergence and application of a modified iteratively regularized Gauss–Newton algorithm

Abstract

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.

A family of preconditioned iteratively regularized methods for nonlinear minimization

“…This source condition was introduced in [KR93] and used in [SRK07] for the analysis of schemes with preconditioning operator L. Hereq is a, possibly nonunique, solution to the noise free equation F (q) = 0. Clearly, for T = I, (1.9) takes the formq…”

“…A discussion of the advantages of (1.9) for convergence, and the associated stopping rule, as compared to other adopted convergence conditions and the Lepskij-type a posteriori stopping rule [L90], [BH05], was presented in [SRK07]. Here, we emphasize that our results extend the methods in [SRK07] to both the use of the more general operator Φ, as well as introducing the use of the inner iterations (1.8) for both T = I and the more general…”

“…and the equivalence between the IRGN presented in [SRK07] for the noise free case in C follows by taking F δ = C − g δ . It was shown in [SRK07] that method (1.3) is very effective for the diffusion optical tomography inverse problem, in which the operator L moves regularization from a B-spline coefficient space directly to the physical space.…”

“…It was shown in [SRK07] that method (1.3) is very effective for the diffusion optical tomography inverse problem, in which the operator L moves regularization from a B-spline coefficient space directly to the physical space. It may also be applied to allow an appropriate weighting on the elements of q (k) to reflect the differing sensitivities of the operator with respect to different physical components.…”

“…

AbstractThe preconditioned iteratively regularized Gauss-Newton algorithm for the minimization of general nonlinear functionals was introduced by Smirnova, Renaut and Khan (2007). In this paper, we establish theoretical convergence results for an extended stabilized family of Generalized Preconditioned Iterative methods which includes M−times iterated Tikhonov regularization with line search.

…”