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In 2021, Z. Fu, Y. Chen and B. Han introduced an inexact Newton regularization (REGINN-IT) using an idea involving the non-stationary iterated Tikhonov regularization scheme for solving nonlinear ill-posed operator equations. In this paper, we suggest a simplified version of the REGINN-IT scheme by using the Bregman distance, duality mapping and a suitable parameter choice strategy to produce an approximate solution. The method is comprised of inner and outer iteration steps. The outer iterates are stopped by a Morozov-type stopping rule, while the inner iterate is executed by making use of the non-stationary iterated Tikhonov scheme. We have studied convergence of the proposed method under some standard assumptions and utilizing tools from convex analysis. The novelty of the method is that it requires computation of the Fréchet derivative only at an initial guess of an exact solution and hence can be identified as more efficient compared to the method given by Z. Fu, Y. Chen and B. Han. Further, in the last section of the paper, we discuss test examples to inspect the proficiency of the method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the simplified Levenberg–Marquardt method for nonlinear ill-posed inverse problems in Hilbert spaces for obtaining stable approximations of solutions to the ill-posed nonlinear equations of the form F ⁢ ( u ) = y {F(u)=y} , where F : 풟 ⁢ ( F ) ⊂ 햴 → 햸 {F:\mathcal{D}(F)\subset\mathsf{U}\to\mathsf{Y}} is a nonlinear operator between Hilbert spaces 햴 {\mathsf{U}} and 햸 {\mathsf{Y}} . The method is defined as follows: u n + 1 δ = u n δ - ( T 0 ∗ ⁢ T 0 + α n ⁢ I ) - 1 ⁢ T 0 ∗ ⁢ ( F ⁢ ( u n δ ) - y δ ) , u_{n+1}^{\delta}=u_{n}^{\delta}-(T_{0}^{\ast}T_{0}+\alpha_{n}I)^{-1}T_{0}^{% \ast}(F(u_{n}^{\delta})-y^{\delta}), where T 0 = F ′ ⁢ ( u 0 ) {T_{0}=F^{\prime}(u_{0})} and T 0 ∗ = F ′ ⁢ ( u 0 ) ∗ {T_{0}^{\ast}=F^{\prime}(u_{0})^{\ast}} . Here F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} denotes the Frèchet derivative of F at an initial guess u 0 ∈ 풟 ⁢ ( F ) {u_{0}\in\mathcal{D}(F)} for the exact solution u † {u^{\dagger}} , F ′ ⁢ ( u 0 ) ∗ {F^{\prime}(u_{0})^{\ast}} is the adjoint of F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} and { α n } {\{\alpha_{n}\}} is an a priori chosen sequence of non-negative real numbers satisfying suitable properties. We use Morozov-type stopping rule to terminate the iterations. Under suitable non-linearity conditions on operator F, we show convergence of the method and also obtain a convergence rate result under a Hölder-type source condition on the element u 0 - u † {u_{0}-u^{\dagger}} . Furthermore, we derive convergence of the method for the case when no source conditions are used and the study concludes with numerical examples which validate the theoretical conclusions. [ABSTRACT FROM AUTHOR]

In 2011, Jin Qinian has proposed a Levenberg–Marquardt method, by making use of duality mapping and the Bregman distance, to get an approximate solution of a nonlinear ill-posed operator equation in Banach space using an a posteriori parameter choice strategy and Morozov-type stopping rule. The method considered by Jin Qinian was an extension of the method proposed by M. Hanke in 1997 for the Hilbert space case. In this paper, we suggest a modified variant of the method, namely, the simplified Levenberg–Marquardt scheme in Banach spaces. The advantage of the method considered in the paper is that, it is also applicable for the operator equation with non-smooth operator. We establish convergence of the method under a modified a posteriori parameter choice strategy which is more feasible than the one considered by Jin Qinian (2011). Numerical example to demonstrate the validity of the considered method is discussed. [ABSTRACT FROM AUTHOR]

In 2010, Qinian Jin considered a regularized Levenberg–Marquardt method in Hilbert spaces for getting stable approximate solution for nonlinear ill-posed operator equation F ⁢ (x) = y , where F : D ⁢ (F) ⊂ X → Y is a nonlinear operator between Hilbert spaces X and Y and obtained rate of convergence results under an appropriate source condition. In this paper, we propose a simplified Levenberg–Marquardt method in Hilbert spaces for solving nonlinear ill-posed equations in which sequence of iteration { x n δ } is defined as x n + 1 δ = x n δ - ( α n ⁢ I + F ′ ⁢ (x 0) * ⁢ F ′ ⁢ (x 0) ) - 1 ⁢ F ′ ⁢ (x 0) * ⁢ (F ⁢ (x n δ) - y δ) . Here { α n } is a decreasing sequence of nonnegative numbers which converges to zero, F ′ ⁢ (x 0) denotes the Fréchet derivative of F at an initial guess x 0 ∈ D ⁢ (F) for the exact solution x † and (F ′ ⁢ (x 0)) * denote the adjoint of F ′ ⁢ (x 0) . In our proposed method, we need to calculate Fréchet derivative of F only at an initial guess x 0 . Hence, it is more economic to use in numerical computations than the Levenberg–Marquardt method used in the literature. We have proved convergence of the method under Morozov-type stopping rule using a general tangential cone condition. In the last section of the paper, numerical examples are presented to demonstrate advantages of the proposed method. [ABSTRACT FROM AUTHOR]

In 2012, Jin Qinian considered an inexact Newton–Landweber iterative method for solving nonlinear ill-posed operator equations in the Banach space setting by making use of duality mapping. The method consists of two steps; the first one is an inner iteration which gives increments by using Landweber iteration, and the second one is an outer iteration which provides increments by using Newton iteration. He has proved a convergence result for the exact data case, and for the perturbed data case, a weak convergence result has been obtained under a Morozov type stopping rule. However, no error bound has been given. In 2013, Kaltenbacher and Tomba have considered the modified version of the Newton–Landweber iterations, in which the combination of the outer Newton loop with an iteratively regularized Landweber iteration has been used. The convergence rate result has been obtained under a Hölder type source condition. In this paper, we study the modified version of inexact Newton–Landweber iteration under the approximate source condition and will obtain an order-optimal error estimate under a suitable choice of stopping rules for the inner and outer iterations. We will also show that the results proved in this paper are more general as compared to the results proved by Kaltenbacher and Tomba in 2013. Also, we will give a numerical example of a parameter identification problem to support our method. [ABSTRACT FROM AUTHOR]

Iteratively regularized Gauss-Newton method considered by Qinian Jin and Min Zhong (2013), where the iterates are defined by convex optimization problem to get the approximate solution of nonlinear ill-posed equation of the form , where is an operator between Banach spaces X and Y, involves calculation of the derivatives of F at each iterate. In this paper, we suggest a modified form of the iteratively regularized Gauss-Newton method in Banach spaces which requires the derivative of F only at an initial approximation of the solution We study convergence analysis of the method under the same a-posteriori rules as considered by Qinian Jin and Min Zhong (2013). The error estimates for this method are obtained under a modified source condition which also involves the derivative of F only at . [ABSTRACT FROM AUTHOR]

In this paper, we study the simplified generalized Gauss–Newton method in a Hilbert scale setting to get an approximate solution of the ill-posed operator equation of the form F ⁢ (x) = y {F(x)=y} where F : D ⁢ (F) ⊆ X → Y {F:D(F)\subseteq X\to Y} is a nonlinear operator between Hilbert spaces X and Y. Under suitable nonlinearly conditions on F, we obtain an order optimal error estimate under the Morozov type stopping rule. [ABSTRACT FROM AUTHOR]

Jin Qinian and Min Zhong [10] considered an iteratively regularized Gauss-Newton method in Banach spaces to find a stable approximate solution of the nonlinear ill-posed operator equation. They have considered a Morozov-type stopping rule (Rule 1) as one of the criterion to stop the iterations and studied the convergence analysis of the method. However, no error estimates have been obtained for this case. In this paper, we consider a modified variant of the method, namely, the simplified Gauss-Newton method under both an a priori as well as a Morozov-type stopping rule. In both cases, we obtain order optimal error estimates under Hölder-type approximate source conditions. An example of a parameter identification problem for which the method can be implemented is discussed in the paper. [ABSTRACT FROM AUTHOR]