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An inexact Newton regularization in Banach spaces based on the nonstationary iterated Tikhonov method. (English) Zbl 1321.65091
Summary: A version of the nonstationary iterated Tikhonov method was recently introduced to regularize linear inverse problems in Banach spaces [Q. Jin and L. Stals, Inverse Probl. 28, No. 10, Article ID 104011, 15 p. (2012; Zbl 1253.47011)]. In the present work we employ this method as inner iteration of the inexact Newton regularization method REGINN [A. Rieder, Inverse Probl. 15, No. 1, 309–327 (1999; Zbl 0969.65049)] which stably solves nonlinear ill-posed problems. Further, we propose and analyze a Kaczmarz version of the new scheme which allows fast solution of problems which can be split into smaller subproblems. As special cases we prove strong convergence of Kaczmarz variants of the Levenberg-Marquardt and the iterated Tikhonov methods in Banach spaces.

MSC:
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
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