We consider a flexible sequential block iterative method for the solution of consistent linear systems of equations and give its convergence analysis. The method is able to use weight matrices and relaxation parameters which can be updated in each iteration whereas the most of previous studies on sequential block iterative methods considered a finite number of weight matrices. Furthermore, we consider the constraint version of the method and give its convergence analysis for the special case of relaxation parameters and weight matrices. We report on some numerical tests with examples taken from the field of image reconstruction from projections. Our numerical results show considerable improvement, specially on noisy data, compared to the other methods which use finite number of weight matrices.

We introduce a new idea of algorithmic structure, called assigning algorithm, using a finite collection of a subclass of strictly quasi-nonexpansive operators. This new algorithm allows the iteration vectors to take steps on a pattern which is based on a connected directed acyclic graph. The sequential, simultaneous, and string-averaging methods for solving convex feasibility problems are the special cases of the new algorithm which may be used to reduce idle time of processors in parallel implementations.We give a convergence analysis for such algorithmic structure with perturbation. Also, we extend some existence results of the split common fixed point problem based on the new algorithm. The performance of the new algorithm is illustrated with numerical examples from computed tomography

We study error minimizing relaxation (EMR) strategies for use in Landweber and Kaczmarz type iterations applied to linear systems with or without convex constraints. Convergence results based on operator theory are given, assuming exact data. The advantages and disadvantages of these relaxation strategies on a noisy and ill-posed problem are illustrated using examples taken from the field of image reconstruction from projections. We also consider combining EMR with penalization.

Column-oriented versions of algebraic iterative methods are interesting alternatives to their row-version counterparts: they converge to a least squares solution, and they provide a basis for saving computational work by skipping small updates. In this paper we consider the case of noise-free data. We present a convergence analysis of the column algorithms, we discuss two techniques (loping and flagging) for reducing the work, and we establish some convergence results for methods that utilize these techniques. The performance of the algorithms is illustrated with numerical examples from computed tomography.

We present convergence analysis of a generalized relaxation of string averaging operators which is based on strictly relaxed cutter operators on a general Hilbert space. In this paper, the string averaging operator is assembled by averaging of strings' endpoints and each string consists of composition of finitely many strictly relaxed cutter operators. We also consider projected version of the generalized relaxation of string averaging operator. To evaluate the study, we recall a wide class of iterative methods for solving linear equations (inequalities) and use the subgradient projection method for solving nonlinear convex feasibility problems.

In this paper, we introduce a subclass of strictly quasi-nonexpansive operators which consists of well-known operators as paracontracting operators (e.g., strictly nonexpansive operators, metric projections, Newton and gradient operators), subgradient projections, a useful part of cutter operators, strictly relaxed cutter operators and locally strongly Féjer operators. The members of this subclass, which can be discontinuous, may be employed by fixed point iteration methods; in particular, iterative methods used in convex feasibility problems. The closedness of this subclass, with respect to composition and convex combination of operators, makes it useful and remarkable. Another advantage with members of this subclass is the possibility to adapt them to handle convex constraints. We give convergence result, under mild conditions, for a perturbation resilient iterative method which is based on an infinite pool of operators in this subclass. The perturbation resilient iterative methods are relevant and important for their possible use in the framework of the recently developed superiorization methodology for constrained minimization problems. To assess the convergence result, the class of operators and the assumed conditions, we illustrate some extensions of existence research works and some new results.

In this paper, we analyze the semiconvergence behavior of a non-stationary sequential block iterative method. Based on a slightly modified problem, in the form of a regularized problem, we obtain three techniques for picking relaxation parameters to control the propagated noise component of the error. Also, we give the convergence analysis of the iterative method using these strategies. The performance of our strategies is shown by examples taken from tomographic imaging.

We analyse a fixed-point iterative method with a finite pool of operators which are subfamily of strictly quasi-nonexpansive operators. These operators, which are not necessarily continuous, may be employed in iterative methods used in convex feasibility problems. Furthermore, members of this subfamily are able to handle convex constraints. The current iterate of the fixed-point iterative method is made by averaging of strings’ endpoints and each string consists of a composition of operators which lie in the pool. To examine the study, we deal with two important pools of operators. The first one is a class of operators which define the algebraic iterative methods, as block iterative projection methods, for solving linear systems of equations (inequalities). The second class consists of the parallel subgradient projection operators for solving nonlinear convex feasibility problems. In both classes, we use optimal relaxation or optimal weight parameters which may break the continuity of the operators used in the classes. The advantages and disadvantages of using these parameters are illustrated using some numerical examples.

Kaczmarz's method - sometimes referred to as the algebraic reconstruction technique - is an iterative method that is widely used in tomographic imaging due to its favorable semi-convergence properties. Specifically, when applied to a problem with noisy data, during the early iterations it converges very quickly toward a good approximation of the exact solution, and thus produces a regularized solution. While this property is generally accepted and utilized, there is surprisingly little theoretical justification for it. The purpose of this paper is to present insight into the semi-convergence of Kaczmarz's method as well as its projected counterpart (and their block versions). To do this we study how the data errors propagate into the iteration vectors and we derive upper bounds for this noise propagation. Our bounds are compared with numerical results obtained from tomographic imaging.

We study a perturbation-resilient iterative method with an infinite pool of operators having a property, which allows using discontinuous operators, weaker than paracontracting. A convergence result is given under an extra condition which is inherently present in some stateof- the-art iterative methods. To evaluate the study, we develop a new algorithmic scheme which generalizes both the string-averaging algorithm and the block-iterative projection methods, and give its convergence analysis for the consistent case.