Hanke–Raus rule for Landweber iteration in Banach spaces

We consider the Landweber iteration for solving linear as well as nonlinear inverse problems in Banach spaces. Based on the discrepancy principle, we propose a heuristic parameter choice rule for choosing the regularization parameter which does not require the information on the noise level, so it i...

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Published in:	Numerische Mathematik Vol. 156; no. 1; pp. 345 - 373
Main Author:	Real, Rommel R.
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2024 Springer Nature B.V
Subjects:	Banach spaces Convergence Inverse problems Iterative methods Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Noise levels Numerical Analysis Numerical and Computational Physics Parameters Regularization Simulation Theoretical 65J20 Numerical solutions of ill-posed problems in abstract spaces regularization
ISSN:	0029-599X, 0945-3245
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Abstract	We consider the Landweber iteration for solving linear as well as nonlinear inverse problems in Banach spaces. Based on the discrepancy principle, we propose a heuristic parameter choice rule for choosing the regularization parameter which does not require the information on the noise level, so it is purely data-driven. According to a famous veto, convergence in the worst-case scenario cannot be expected in general. However, by imposing certain conditions on the noisy data, we establish a new convergence result which, in addition, requires neither the Gâteaux differentiability of the forward operator nor the reflexivity of the image space. Therefore, we also expand the applied range of the Landweber iteration to cover non-smooth ill-posed inverse problems and to handle the situation that the data is contaminated by various types of noise. Numerical simulations are also reported.
AbstractList	We consider the Landweber iteration for solving linear as well as nonlinear inverse problems in Banach spaces. Based on the discrepancy principle, we propose a heuristic parameter choice rule for choosing the regularization parameter which does not require the information on the noise level, so it is purely data-driven. According to a famous veto, convergence in the worst-case scenario cannot be expected in general. However, by imposing certain conditions on the noisy data, we establish a new convergence result which, in addition, requires neither the Gâteaux differentiability of the forward operator nor the reflexivity of the image space. Therefore, we also expand the applied range of the Landweber iteration to cover non-smooth ill-posed inverse problems and to handle the situation that the data is contaminated by various types of noise. Numerical simulations are also reported.
Author	Real, Rommel R.
Author_xml	– sequence: 1 givenname: Rommel R. surname: Real fullname: Real, Rommel R. email: rrreal@up.edu.ph organization: Department of Mathematics, Physics, and Computer Science, University of the Philippines Mindanao
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Fields201881247276381087610.3934/mcrf.2018011 ClasonCL∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty $$\end{document} fitting for inverse problems with uniform noiseInverse Probl.20122810298790210.1088/0266-5611/28/10/104007 ClasonCNhuVHBouligand–Landweber iteration for a non-smooth ill-posed problemNumer. Math.20191424789832397585010.1007/s00211-019-01038-6 JinQInexact Newton-Landweber iteration in Banach spaces with nonsmooth convex penalty termsSIAM J. Numer. Anal.201553523892413341446910.1137/130940505 Jin, Q., Wang, W.: Analysis of the iteratively regularized Gauss–Newton method under a heuristic rule. Inverse Probl. 34(3) (2018) ZhongMWangWJinQRegularization of inverse problems by two-point gradient methods in Banach spacesNumer. 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Inverse Probl. 32(8) (2016) JinQWangWLandweber iteration of Kaczmarz type with general non-smooth convex penalty functionalsInverse Probl.2013298122308468510.1088/0266-5611/29/8/085011 RappazJApproximation of a nondifferentiable nonlinear problem related to MHD equilibriaNumer. 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References_xml	– reference: HankeMNeubauerAScherzerOA convergence analysis of the Landweber iteration for nonlinear ill-posed problemsNumer. Math.19957212137135970610.1007/s002110050158 – reference: ChristofCMeyerCWaltherSClasonCOptimal control of a non-smooth semilinear elliptic equationMath. Control Relat. Fields201881247276381087610.3934/mcrf.2018011 – reference: RudinLIOsherSFatemiENonlinear total variation based noise removal algorithmsPhys. D Nonlinear Phenom.1992601–4259268336340110.1016/0167-2789(92)90242-F – reference: SchusterTKaltenbacherBHofmannBKazimierskiKRegularization Methods in Banach Spaces2012BerlinWalter de Gruyter GmbH Co.KG10.1515/9783110255720 – reference: Jin, Q.: Hanke–Raus heuristic rule for variational regularization in Banach spaces. Inverse Probl. 32(8) (2016) – reference: BakushinskiiABRemarks on choosing a regularization parameter using the quasioptimality and ratio criterionUSSR Comput. Math. Math. Phys.198424418118210.1016/0041-5553(84)90253-2 – reference: ClasonCNhuVHBouligand–Landweber iteration for a non-smooth ill-posed problemNumer. Math.20191424789832397585010.1007/s00211-019-01038-6 – reference: JinQOn a regularized Levenberg–Marquardt method for solving nonlinear inverse problemsNumer. Math.20101152229259260696110.1007/s00211-009-0275-x – reference: Tröltzsch, F.: Optimal Control of Partial Differential Equations, vol. 112. Graduate Studies in Mathematics. Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels. American Mathematical Society, Providence, RI, pp. xvi+399 (2010) – reference: JinQInexact Newton-Landweber iteration in Banach spaces with nonsmooth convex penalty termsSIAM J. Numer. Anal.201553523892413341446910.1137/130940505 – reference: Boţ, R.I., Hein, T.: Iterative regularization with a geeral penalty term—theory and applications to L1 and TV regularization. Inverse Probl. 28 (2012) – reference: HankeMRausTA general heuristic for choosing the regularization parameter in ill-posed problemsSIAM J. Sci. Comput.1996174956972139535810.1137/0917062 – reference: Kaltenbacher, B., Schöpfer, F., Schuster, T.: Iterative methods for nonlinear inverse ill-posed problems in Banach spaces: convergence and applications to parameter identification problems. Inverse Probl. 25(6) (2009) – reference: Schöpfer, F., Louis, A.K., Schuster, T.: Nonlinear iterative methods for linear ill-posed problems in Banach spaces. Inverse Probl. 22(1) (2006) – reference: RappazJApproximation of a nondifferentiable nonlinear problem related to MHD equilibriaNumer. Math.198445111713376188410.1007/BF01379665 – reference: Jin, Q., Wang, W.: Analysis of the iteratively regularized Gauss–Newton method under a heuristic rule. Inverse Probl. 34(3) (2018) – reference: RealRJinQA revisit on Landweber iterationInverse Probl.202036775011412132310.1088/1361-6420/ab8bc4 – reference: ZhangZJinQHeuristic rule for non-stationary iterated Tikhonov regularization in Banach spacesInverse Probl.20183411385608810.1088/1361-6420/aad918 – reference: JinQOn a heuristic stopping rule for the regularization of inverse problems by the augmented Lagrangian methodNumer. Math.20161364973992367159410.1007/s00211-016-0860-8 – reference: Liu, H., Real, R., Lu, X., Jia, X., Jin, Q.: Heuristic discrepancy principle for variational regularization of inverse problems. Inverse Probl. 36(7) (2020) – reference: FuZJinQZhangZHanBChenYAnalysis of a heuristic rule for the IRGNM in Banach spaces with convex regularization termsInverse Probl.202036775002412131410.1088/1361-6420/ab8448 – reference: JinQWangWLandweber iteration of Kaczmarz type with general non-smooth convex penalty functionalsInverse Probl.2013298122308468510.1088/0266-5611/29/8/085011 – reference: HubmerSSherinaEKindermannSRaikKA numerical comparison of some heuristic stopping rules for nonlinear Landweber iterationElectron. Trans. Numer. Anal.202257216241451652210.1553/etna_vol57s216 – reference: ZǎlinescuCConvex Analysis in General Vector Spaces2002River Edge, NJWorld Scientific10.1142/5021 – reference: CioranescuIGeometry of Banach Spaces, Duality Mappings, and Nonlinear Problems1990DordrechtKluwer10.1007/978-94-009-2121-4 – reference: Temam, R.: A non-linear eigenvalue problem: the shape at equilibrium of a confined plasma. Arch. Ration. Mech. Anal. 60(1), 51–73 (1975/76) – reference: ZhongMWangWJinQRegularization of inverse problems by two-point gradient methods in Banach spacesNumer. Math.20191433713747402066910.1007/s00211-019-01068-0 – reference: Zhu, M., Chan, T.: An efficient primal-dual hybrid gradient algorithm for total variation image restoration. In: UCLA CAM Report (May 2008) – reference: AsterRCBorchersBThurberCHParameter Estimation and Inverse Problems2005AmsterdamElsevier Academic Press10.1016/S0074-6142(05)80014-2 – reference: ClasonCL∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty $$\end{document} fitting for inverse problems with uniform noiseInverse Probl.20122810298790210.1088/0266-5611/28/10/104007 – reference: JinBLorenzDAHeuristic parameter-choice rules for convex variational regularization based on error estimatesSIAM J. Numer. Anal.201048312081229267957810.1137/100784369 – reference: JinQLandweber–Kaczmarz method in Banach spaces with inexact inner solversInverse Probl.20163210362702110.1088/0266-5611/32/10/104005 – reference: KikuchiFNakazatoKUshijimaTFinite element approximation of a nonlinear eigenvalue problem related to MHD equilibriaJpn. J. Appl. Math.19841236940384080310.1007/BF03167065 – ident: 1389_CR22 doi: 10.1088/1361-6420/ab844a – volume: 45 start-page: 117 issue: 1 year: 1984 ident: 1389_CR23 publication-title: Numer. Math. doi: 10.1007/BF01379665 – volume-title: Convex Analysis in General Vector Spaces year: 2002 ident: 1389_CR33 doi: 10.1142/5021 – volume: 24 start-page: 181 issue: 4 year: 1984 ident: 1389_CR2 publication-title: USSR Comput. Math. Math. Phys. doi: 10.1016/0041-5553(84)90253-2 – volume: 17 start-page: 956 issue: 4 year: 1996 ident: 1389_CR10 publication-title: SIAM J. Sci. 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Snippet	We consider the Landweber iteration for solving linear as well as nonlinear inverse problems in Banach spaces. Based on the discrepancy principle, we propose a...
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SubjectTerms	Banach spaces Convergence Inverse problems Iterative methods Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Noise levels Numerical Analysis Numerical and Computational Physics Parameters Regularization Simulation Theoretical
Title	Hanke–Raus rule for Landweber iteration in Banach spaces
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