On convergence rates for iteratively regularized procedures with linear penalty terms

Abstract

The impact of this paper is twofold. First, we study convergence rates of the iteratively regularized Gauss-Newton (IRGN) algorithm with a linear penalty term under a generalized source assumption and show how the regularizing properties of new iterations depend on the solution smoothness. Secondly, we introduce an adaptive IRGN procedure, which is investigated under a relaxed smoothness condition. The introduction and analysis of a more general penalty term are of great importance since, apart from bringing stability to the numerical scheme designed for solving a large class of applied inverse problems, it allows us to incorporate various types of a priori information available on the model. Both a priori and a posteriori stopping rules are investigated. For the a priori stopping rule, optimal convergence rates are derived. A numerical example illustrating convergence rates is considered.

A nonstandard approximation of pseudoinverse and a new stopping criterion for iterative regularization

“…Numerical experiments, for the most part, con rm the theoretical ndings (see [19,34] for details), which means that the special structure, imposed by a source-type condition, is essential for the behavior of the sequence { }. At the same time, practical implementation of the iterative process (1.5) with various choices of suggests that the requirement for ‖ ‖ in (1.16) to be small is not that critical and can possibly be relaxed.…”

On iteratively regularized predictor–corrector algorithm for parameter identification *

“…Regardless of a particular disease, fitting model predictions for an invading pathogen to reported incidence series yields an ill-posed problem due to excessive noise propagation coupled with unavoidable delays in processing of epidemic data. In order to solve this ill-posed problem in a stable manner, regularized Gauss-Newton or Levenberg-Marquardt algorithms [3,4,14,18,22,26,27,29,32] are commonly used to minimize the cost functional. Oftentimes, the biological model (which may be a system of nonlinear ordinary or partial differential equations), constraining the function minimization problem, does not have a closed form-solution and has to be solved numerically at every step of the iterative process.…”

“…It employs a predictor-corrector kind of algorithm, where one updates θ while freezing u, and then u is modified while θ is kept unchanged. More specifically, given θ k u k , one transitions from θ k to θ k+1 by applying one step of the modified iteratively regularized Gauss-Newton (MIRGN) procedure [3,4,18,26,27]:…”