Abstract. In this work we present and analyze a Kaczmarz version of the iterative regularization scheme REGINN-Landweber for nonlinear ill-posed problems in Banach spaces [Jin, Inverse Problems 28(2012), 065002]. Kaczmarz methods are designed for problems which split into smaller subproblems which are then processed cyclically during each iteration step. Under standard assumptions on the Banach space and on the nonlinearity we prove stability and (norm-)convergence as the noise level tends to zero. Further, we test our scheme on the inverse problem of 2D electric impedance tomography not only to illustrate our theoretical findings but also to study the influence of different Banach spaces on the reconstructed conductivities.

A version of the nonstationary iterated Tikhonov method was recently introduced to regularize linear inverse problems in Banach spaces [7]. In the present work we employ this method as inner iteration of the inexact Newton regularization method REGINN [14] 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.

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In this article we propose a novel strategy for choosing the Lagrange multipliers in the Levenberg–Marquardt method for solving ill-posed problems modeled by nonlinear operators acting between Hilbert spaces. Convergence analysis results are established for the proposed method, including monotonicity of iteration error, geometrical decay of the residual, convergence for exact data, stability and semi-convergence for noisy data. Numerical experiments are presented for an elliptic parameter identification two-dimensional electrical impedance tomography problem. The performance of our strategy is compared with standard implementations of the Levenberg–Marquardt method (using a priori choice of the multipliers).

In this paper we investigate convergence and stability properties of an adaptation to Banach spaces of the algorithm REGINN (Rieder 1999 Inverse Problems 15 309-27). This inexact Newton method solves nonlinear inverse problems by means of linearizing the equation around the current iterate and subsequently applying a regularization technique in the so-called inner iteration in order to obtain a stable approximation of the resulting linearized system. The current iterate is then updated by adding this approximate solution.Using a generic gradient-like method as inner iteration, we provide a whole converge analysis of this Newton-type method, proving, under reasonable assumptions, strong convergence of the generated sequence to a solution of the inverse problem in the noiseless situation and the regularization property in the noisy-data case.Some numerical experiments are performed at the end of the paper for providing the necessary support to the theoretical results