Convergence rates for iteratively regularized Gauss–Newton method subject to stability constraints

Self Cite

In this paper, we study the nonstationary iterated Tikhonov regularization method (NITRM) proposed by Jin et al. [19] to solve the inverse problems, where the inverse mapping fulfills a Holder stability estimate. The iterates of NITRM are defined through certain minimization problems in the settings of Banach spaces. In order to study the various important characteristics of the sought solution, we consider the non-smooth uniformly convex penalty terms in the minimization problems. In the case of noisy data, we terminate the method via a discrepancy principle and show the strong convergence of the iterates as well as the convergence with respect to the Bregman distance. For noise free data, we show the convergence of the iterates to the sought solution. Additionally, we derive the convergence rates of NITRM method for both the noisy and noise free data that are missing from the literature. In order to derive the convergence rates, we solely utilize the Holder stability of the inverse mapping that opposes the standard analysis which requires a source condition as well as a nonlinearity estimate to be satisfied by the inverse mapping. Finally, we discuss three numerical examples to show the validity of our results.

Section: Advantages Of This Workmentioning

confidence: 99%

“…The motivation for this work comes from the notable works of [8,9]. We also refer to [21][22][23]28] for some recent works that incorporated stability estimates to study the convergence analysis of various iterative regularization methods.…”

Section: Introductionmentioning

confidence: 99%

Nonstationary iterated Tikhonov regularization: convergence analysis via Hölder stability

Mittal¹,

Giri²

2022

Self Cite

“…Here, we extend the convex functional to the general convex term θ. We thus modify (49) into the form (48), whose analog has already been introduced in [28].…”

Section: Convergence and Convergence Rate In Banach Spacesmentioning

confidence: 99%

“…Motivated by [25], the inexact Newton-Landweber (6) for the exact data was analyzed in [16]. And then, literature [27,28] used the Hölder stability estimate to study the Gauss-Newton method and iteratively regularized Landweber iteration, respectively.…”

Section: Introductionmentioning

confidence: 99%

Convergence analysis of inexact Newton–Landweber iteration under Hölder stability

Xia¹,

Han²,

Fu³

2022

In this paper, we focus on a class of inverse problems with Lipschitz continuous Fréchet derivatives both in Hilbert spaces and Banach spaces. The convergence and convergence rate of the inexact Newton−Landweber method (INLM) for such problems are presented under some assumptions. For the inverse problems in Hilbert spaces, we revisit the convergence result and the convergence rate of the INLM under Lipschitz condition and Hölder stability. Furthermore, the INLM for nonlinear inverse problems in Banach spaces is also considered. By using a Hölder stability corresponding to the Bregman distance, we derive the convergence property and convergence rate of the method.

“…In [11], the authors considered the stable solution of nonlinear inverse problems satisfying a conditional stability estimate by Tikhonov regularization in Hilbert scales. The latest related works were [29,30], based on the conditional stability, the convergence rate of Landweber-typed iteration, iteratively regularized Landweber iteration and iteratively regularized Gauss-Newton method in Banach spaces were discussed, respectively.…”

Section: Introductionmentioning

confidence: 99%

An asymptotical regularization with convex constraints for inverse problems

Zhong¹,

Wang²,

Tong³

2022

We investigate the method of asymptotical regularization for the stable approximate solution of nonlinear ill-posed problems $F(x)=y$ in Hilbert spaces. The method consists of two components, an outer Newton iteration and an inner scheme providing increments by solving a local coupling linearized evolution equations. In addition, a non-smooth uniformly convex functional has been embedded in the evolution equations which is allowed to be non-smooth, including $L^1$-liked and total variation-like penalty terms. We establish convergence properties of the method, derive stability estimates, and perform the convergence rate under the H\"{o}lder continuity of the inverse mapping. Furthermore, based on Runge-Kutta (RK) discretization, different kinds of iteration schemes can be developed for numerical realization. In our numerical experiments, four types iterative scheme, including Landweber type, $1$-stage explicit, implicit Euler and $2$-stage RK are presented to illustrate the performance of the proposed method.