In this article we propose and analyze a Levenberg–Marquardt–Kaczmarz-type (LMK) method for obtaining stable approximate solutions to systems of ill-posed equations modeled by non-linear operators acting between Hilbert spaces. We extend to the LMK iteration the strategy proposed in [A. Leitão, F. Margotti and B. F. Svaiter, Range-relaxed criteria for choosing the Lagrange multipliers in the Levenberg–Marquardt method, IMA J. Numer. Anal. 41 2021, 4, 2962–2989] for choosing the Lagrange multipliers in the Levenberg–Marquardt (LM) method. Our main goal is to devise a simple (and easy to implement) strategy for computing the multiplier in each iterative step, such that the resulting LMK iteration is both stable and numerically efficient. Convergence analysis for the proposed LMK type method is provided, including convergence for exact data, stability and semi-convergence. Numerical experiments using real data are presented for a 2D parameter identification problem, namely the Electrical Impedance Tomography (EIT) problem. The mathematical model known as complete electrode model (EIT-CEM) is considered. The obtained numerical results validate the efficiency of the proposed LMK-type method.

In this article we propose and analyze a nonstationary iterated Tikhonov Kaczmarz (iTK) type method for obtaining stable approximate solutions to systems of ill-posed equations modeled by linear operators acting between Hilbert spaces. We generalize for the iTK iteration the criteria proposed in [5] for the iterated Tikhonov method. The goal is to devise an efficient strategy for choosing the Lagrange multipliers in this method. Convergence analysis for the resulting iTK method is provided, including convergence for exact data, stability and semi-convergence. Numerical experiments are presented for two distinct applications, namely: an image deblurring problem and a 2D elliptic parameter identification problem (the inverse potential problem). The obtained numerical results validate the efficiency of the proposed method.

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In this study, we review, compare, analyze and propose new variants of first-order methods for the convex hull membership problem (CHMP). Though CHMP can be formulated as a linear or a quadratic programming problem which could then be tackled with standard first-orders methods, here we focus on a geometric algorithm, introduced recently in [17], called Triangle Algorithm (TA). We formally establish TA connections with the well-known conditional gradient (Frank-Wolfe) methods. Despite that TA has its foundation on a theorem of alternative, known as distance duality, we show that its iteration mechanism is quite similar to a Frank-Wolfe iteration. This new point of view allows us to devise variants of the Triangle and Frank-Wolfe algorithms showing promising results in practice. Furthermore, based on the distance duality theorem, we develop appropriate stopping criteria for first-order methods such as conditional gradient and projected gradient applied to the CHMP. Our numerical experiments on random instances of CHMP indicate that an Away-Step version of Frank-Wolfe achieves the best performance in comparison to other first-order methods considered in this paper. We also evaluate the performance of such algorithms when applied to linear programming feasibility problems, which under mild assumptions can be cast as a CHMP.

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