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This paper investigates the inverse problem of determining a heat source in the parabolic heat equation using the usual conditions. Firstly, the problem is reduced to an equivalent problem which is easy to handle using variational iteration method. Secondly, variational iteration method is used to solve the reduced problem. Using this method a rapid convergent sequence can be produced which tends to the exact solution of the problem. Furthermore, variational iteration method does not require the discretization of the problem. Two numerical examples are presented to illustrate the strength of the method. [ABSTRACT FROM AUTHOR]

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 [ABSTRACT FROM AUTHOR]

Abstract  We develop a general convergence analysis for a class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients. [ABSTRACT FROM AUTHOR]

We consider the methods xδ=xnδ-gαn for solving nonlinear ill-posed inverse problems F(x) = y using the only available noise data yδsatisfying ||Yδ - y|| ≤ δ with a given small noise level δ > 0. We terminate the iteration by the discrepancy principle Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed., 0 ≤ n < nδ, with a given number τ > 1. Under certain conditions on {αn} and F, we prove for a large class of spectral filter functions {gα} the convergence of Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. to a true solution as δ → 0. Moreover, we derive the order optimal rates of convergence when certain Hölder source conditions hold. Numerical examples are given to test the theoretical results. [ABSTRACT FROM AUTHOR]

The article discusses a study which explores the strong convergence of an iterative method for inverse strongly accretive operators in the framework of Banach spaces. The study found that the variational inequality for an inverse strongly monotone operator in the framework of Banach spaces is equivalent to some fixed-point problem. The study also assessed that the strong convergence for an iterative method for inverse strongly accretive operator is based on that of the inverse strongly monotone operator.