Iteratively regularized Landweber iteration method: Convergence analysis via Hölder stability

Mittal, Gaurav; Giri, Ankik Kumar

doi:10.1016/j.amc.2020.125744

Cited by 17 publications

(8 citation statements)

References 25 publications

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“…(2) For IRLIM, it is known that in the presence of noisy data, iterates remain in a ball around the initial guess but no convergence to the exact solution is known whenever noise tends to 0. Consequently, the results of this paper are better than that of [21]. In order to compare the convergence rates of IRLIM with NITRM, suppose φ 1 (n) ≈ Cn for some C > 0.…”

Section: Advantages Of This Workmentioning

confidence: 86%

“…So, this paper fills a research gap in the literature related to deriving the convergence rates of NITRM. (b) The convergence rates of various iterative methods such as Landweber iteration method (LIM) [8], inexact Newton-Landweber iteration method (INLIM) [28], iteratively regularized Landweber iteration method (IRLIM) [21], projected steepest descent method (PSDM) [9], iteratively regularized Gauss-Newton method (IRGNM) [22] and NITRM through HSE (3.5) are given in table 1. In table 1, we use CR as an abbreviation for convergence rates.…”

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%

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Nonstationary iterated Tikhonov regularization: convergence analysis via Hölder stability

Mittal¹,

Giri²

2022

Inverse Problems

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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.

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Section: Advantages Of This Workmentioning

confidence: 86%

Section: Advantages Of This Workmentioning

confidence: 99%

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Nonstationary iterated Tikhonov regularization: convergence analysis via Hölder stability

Mittal¹,

Giri²

2022

Inverse Problems

Self Cite

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“…These conditions have previously been used in [25] for the convergence analysis of the nonlinear Landweber iteration. One can consult [25,27] for several examples satisfying assumption 3.1.…”

Section: Convergence and Convergence Rate In Hilbert 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

Inverse Problems

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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.

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Infinite-Dimensional Inverse Problems with Finite Measurements

Alberti

Santacesaria

2021

Arch Rational Mech Anal

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Iteratively regularized Landweber iteration method: Convergence analysis via Hölder stability

Cited by 17 publications

References 25 publications

Nonstationary iterated Tikhonov regularization: convergence analysis via Hölder stability

Nonstationary iterated Tikhonov regularization: convergence analysis via Hölder stability

Convergence analysis of inexact Newton–Landweber iteration under Hölder stability

Infinite-Dimensional Inverse Problems with Finite Measurements

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