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This paper considers different versions of block-column iterative (BCI) methods for solving nonnegative constrained linear least squares problems. We present the convergence analysis for a family of stationary BCI methods with nonnegativity constraints (BCI-NC), which is applicable to linear complementarity problems (LCP). We also consider the flagging idea for BCI methods, which allows saving computational work by skipping small updates. Also, we combine the BCI-NC algorithm and the flagging version of a nonstationary BCI method with nonnegativity constraints to derive a convergence analysis for the resulting method (BCI-NF). The performance of our algorithms is shown on ill-posed inverse problems taken from tomographic imaging. We compare the BCI-NF and BCI-NC algorithms with three recent algorithms: the inner-outer modulus method (Modulus-CG method), the modulus-based iterative method to Tikhonov regularization with nonnegativity constraint (Mod-TRN method), and nonnegative flexible CGLS (NN-FCGLS) method. Our algorithms are able to produce more stable results than the mentioned methods with competitive computational times.

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.

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.

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 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 state-of-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.

An algorithm for solving convex feasibility problem for a finite family of convex sets is considered. The acceleration scheme of De Pierro (em Methodos de projecao para a resolucao de sistemas gerais de equacoes algebricas lineares. Thesis (tese de Doutoramento), Instituto de Matematica da UFRJ, Cidade Universitaria, Rio de Janeiro, Brasil, 1981), which is designed for simultaneous algorithms, is used in the algorithm to speed up the fully sequential cyclic subgradient projections method. A convergence proof is presented. The advantage of using this strategy is demonstrated with some examples.

We give a detailed study of the semiconvergence behavior of projected nonstationary simultaneous iterative reconstruction technique (SIRT) algorithms, including the projected Landweber algorithm. We also consider the use of a relaxation parameter strategy, proposed recently for the standard algorithms, for controlling the semiconvergence of the projected algorithms. We demonstrate the semiconvergence and the performance of our strategies by examples taken from tomographic imaging. Funding Agencies|Danish Research Council for Technology and Production Sciences|274-07-0065

We propose and study a block-iterative projection method for solving linear equations and/or inequalities. The method allows diagonal componentwise relaxation in conjunction with orthogonal projections onto the individual hyperplanes of the system, and is thus called diagonally relaxed orthogonal projections (DROP). Diagonal relaxation has proven useful in accelerating the initial convergence of simultaneous and block-iterative projection algorithms, but until now it was available only in conjunction with generalized oblique projections in which there is a special relation between the weighting and the oblique projections. DROP has been used by practitioners, and in this paper a contribution to its convergence theory is provided. The mathematical analysis is complemented by some experiments in image reconstruction from projections which illustrate the performance of DROP.

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 class of block-iterative (BI) methods proposed in image reconstruction for solving linear systems. A subclass, symmetric block-iteration (SBI), is derived such that for this subclass both semi-convergence analysis and stopping-rules developed for fully simultaneous iteration apply. Also results on asymptotic convergence are given, e.g., BI exhibit cyclic convergence irrespective of the consistency of the linear system. Further it is shown that the limit points of SBI satisfy a weighted least-squares problem. We also present numerical results obtained using a trained stopping rule on SBI.