Parallel projected aggregation methods for solving large inconsistent systems

The Projected Aggregation Methods (PAM) for solving linear systems of equali- ties and/or inequalities, generate a new iterate xk+1 by projecting the current point xk onto a separating hyperplane generated by a given linear combination of the original hyperplanes and/or halfspaces. In [9, 16, 17, 18] we introduced acceleration schemes for solving linear systems within a PAM like framework. The basic idea was to force the next iterate to belong to the convex region de ned by the new separating/or aggregated hyperplane computed in the previous iteration. In this paper we extend the above mentioned methods to the problem of nding the least squares solution to inconsistent systems. In the new algorithm we used a scheme of incomplete alternate projections for minimizing the proximity function, similar to the one of Csisz ar y Tusn ady described in [4] which uses exact projections. The parallel simultaneous projection ACCIM algorithm in [16] is very eÆcient for ob- taining approximations with suitable properties, and is the basis for calculating the incomplete intermediate projections. We discuss the convergence properties of the new algorithm and also present numerical experiences obtained by applying it to image reconstruction problems using the SNARK93 system [3].Eje: Teoría (TEOR)Red de Universidades con Carreras en Informática (RedUNCI

The parallel surrogate constraint approach to the linear feasibility problem

The linear feasibility problem arises in several areas of applied mathematics and medical science, in several forms of image reconstruction problems. The surrogate constraint algorithm of Yang and Murty for the linear feasibility problem is implemented and analyzed. The sequential approach considers projections one at a time. In the parallel approach, several projections are made simultaneously and their convex combination is taken to be used at the next iteration. The sequential method is compared with the parallel method for varied numbers of processors. Two improvement schemes for the parallel method are proposed and tested. © Springer-Verlag Berlin Heidelberg 1996

An incomplete projections algorithm for solving large inconsistent linear systems

The Projected Aggregation Methods (PAM) for solving linear systems of equalities and/or inequalities, generate a new iterate x k+1 by projecting the current point x k onto a separating hyperplane generated by a given linear combination of the original hyperplanes and/or halfspaces. The authors have introduced in several papers new acceleration schemes for solving systems of linear equations and inequalities respectively, within a PAM like framework. The basic idea was to force the next iterate to belong to the convex region defined by the new separating or aggregated hyperplane computed in the previous iteration. In this paper the above mentioned methods are extended to the problem of finding the least squares solution to inconsistent systems. The new algorithm is based upon a new scheme of incomplete alternate projections for minimizing the proximity function. The parallel simultaneous projections ACCIM algorithm, published by the authors, is the basis for calculating the incomplete intermediate projections. The convergence properties of the new algorithm are given together with numerical experiences obtained by applying it to image reconstruction problems using the SNARKQ3 system.Facultad de Ciencias Exacta

Parallel projected aggregation methods for solving large inconsistent systems

The Projected Aggregation Methods (PAM) for solving linear systems of equali- ties and/or inequalities, generate a new iterate xk+1 by projecting the current point xk onto a separating hyperplane generated by a given linear combination of the original hyperplanes and/or halfspaces. In [9, 16, 17, 18] we introduced acceleration schemes for solving linear systems within a PAM like framework. The basic idea was to force the next iterate to belong to the convex region de ned by the new separating/or aggregated hyperplane computed in the previous iteration. In this paper we extend the above mentioned methods to the problem of nding the least squares solution to inconsistent systems. In the new algorithm we used a scheme of incomplete alternate projections for minimizing the proximity function, similar to the one of Csisz ar y Tusn ady described in [4] which uses exact projections. The parallel simultaneous projection ACCIM algorithm in [16] is very eÆcient for ob- taining approximations with suitable properties, and is the basis for calculating the incomplete intermediate projections. We discuss the convergence properties of the new algorithm and also present numerical experiences obtained by applying it to image reconstruction problems using the SNARK93 system [3].Eje: Teoría (TEOR)Red de Universidades con Carreras en Informática (RedUNCI

An acceleration scheme for solving convex feasibility problems using incomplete projection algorithms

The Projected Aggregation Methods (PAM) for solving linear systems of equalities and/or inequalities, generate a new iterate xk+1by projecting the current point zk onto a separating hyperplane generated by a given linear combination of the original hyperplanes or halfspaces. In H. Scolnik et al we introduced acceleration schemes for solving systems of linear equations by applying optimization techniques to the problem of finding the optimal combination of the hyperplanes within a PAM like framework. In this paper we generalize those results, introducing a new accelerated iterative method for solving systems of linear inequalities, together with the corresponding theoretical convergence results. In order to test its efficiency, numerical results obtained applying the new acceleration scheme to two algorithms introduced by U. M. García-Palomares and F. J. González-Castaño are given.Material digitalizado en SEDICI gracias a la Biblioteca de la Facultad de Ingeniería (UNLP).Facultad de Ciencias Exacta

Incomplete oblique projections method for solving regularized least-squares problems in image reconstruction

In this paper we improve on the incomplete oblique projections (IOP) method introduced previously by the authors for solving inconsistent linear systems, when applied to image reconstruction problems. That method uses IOP onto the set of solutions of the augmented system Ax - r = b, and converges to a weighted least-squares solution of the system Ax=b. In image reconstruction problems, systems are usually inconsistent and very often rank-deficient because of the underlying discretized model. Here we have considered a regularized least-squares objective function that can be used in many ways such as incorporating blobs or nearest-neighbor interactions among adjacent pixels, aiming at smoothing the image. Thus, the oblique incomplete projections algorithm has been modified for solving this regularized model. The theoretical properties of the new algorithm are analyzed and numerical experiments are presented showing that the new approach improves the quality of the reconstructed images.Material digitalizado en SEDICI gracias a la Biblioteca de la Facultad de Ingeniería (UNLP).Facultad de Ciencias Exacta

Algorithms for linear and convex feasibility problems: A brief study of iterative projection, localization and subgradient methods

Ankara : Department of Industrial Engineering and Institute of Engineering and Sciences, Bilkent Univ., 1998.Thesis (Ph.D.) -- Bilkent University, 1998.Includes bibliographical references leaves 86-93.Several algorithms for the feasibility problem are investigated. For linear systems, a number of different block projections approaches have been implemented and compared. The parallel algorithm of Yang and Murty is observed to be much slower than its sequential counterpart. Modification of the step size has allowed us to obtain a much better algorithm, exhibiting considerable speedup when compared to the sequential algorithm. For the convex feasibility problem an approach combining rectangular cutting planes and subgradients is developed. Theoretical convergence results are established for both ca^es. Two broad classes of image recovery problems are formulated as linear feasibility problems and successfully solved with the algorithms developed.Özaktaş, HakanPh.D