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Acceleration scheme for parallel projected aggregation methods for solving large linear systems

ANNALS OF OPERATIONS RESEARCH 2002年第1-4期117卷 95-115页

作者： Scolnik, H Echebest, N Guardarucci, MT Vacchino, MC Univ Buenos Aires Fac Ciencias Exactas & Nat Dept Comp RA-1053 Buenos Aires DF Argentina Univ Nacl La Plata Fac Ciencias Exactas Dept Matemat La Plata Argentina

The projected aggregation methods generate the new point (k+)1 as the projection of x(k) onto an "aggregate" hyperplane usually arising from linear combinations of the hyperplanes defined by the blocks. The aim of this paper is to improve the speed of convergence of a particular kind of them by projecting the directions given by the blocks onto the aggregate hyperplane defined in the last iteration. For that purpose we apply the scheme introduced in "A new method for solving large sparse systems of linear equations using row projections" [11], for a given block projection algorithm, to some new methods here introduced whose main features are related to the fact that the projections do not need to be accurately computed. Adaptative splitting schemes are applied which take into account the structure and conditioning of the matrix. It is proved that these new highly parallel algorithms improve the original convergence rate and present numerical results which show their computational efficiency.

关键词： projected aggregation methods row partition strategies parallel iterative methods

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PARALLEL projected aggregation methods FOR SOLVING THE CONVEX FEASIBILITY PROBLEM

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SIAM JOURNAL ON OPTIMIZATION 1993年第4期3卷 882-900页

作者： Garcia-Palomares, Ubaldo Univ Simon Bolivar Dept Proc & Sistemas Apartado 89000 Caracas 1086 A Venezuela

Convergence conditions are established for new sequential and parallel projected aggregation methods (PAMs) that find a feasible point of a large system of convex inequalities and linear equations. To formulate a multiprocessor method suitable for solving a nonstructured convex system, block iterative methods are used and all system constraints are simultaneously processed. Each processor is assigned the task of finding closer points to one block subsystem, so that at every iteration each processor proposes a point closer (in some norm) to a group of the system constraints, and a head processor combines the proposals and generates a point closer to the original system. These parallel versions appear amenable to multiprocessing. Numerical results are reported that give hints on how to code these methods in a multiprocessor environment.

关键词： convex systems convex feasibility problem parallel processing projected aggregation methods

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An accelerated iterative method with diagonally scaled oblique projections for solving linear feasibility problems

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ANNALS OF OPERATIONS RESEARCH 2005年第1期138卷 235-257页

作者： Echebest, N Guardarucci, MT Scolnik, HD Vacchino, MC Univ Buenos Aires Fac Ciencias Exactas & Nat Dept Computac RA-1053 Buenos Aires DF Argentina Univ Nacl La Plata Fac Ciencias Exactas Dept Matemat La Plata Buenos Aires Argentina

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 or halfspaces. In Scolnik et al. ( 2001, 2002a) and Echebest et al. ( 2004) acceleration schemes for solving systems of linear equations and inequalities respectively were introduced, within a PAM like framework. In this paper we apply those schemes in an algorithm based on oblique projections reflecting the sparsity of the matrix of the linear system to be solved. We present the corresponding theoretical convergence results which are a generalization of those given in Echebest et al. ( 2004). We also present the numerical results obtained applying the new scheme to two algorithms introduced by Garcia-Palomares and Gonzalez-Castano ( 1998) and also the comparison of its efficiency with that of Censor and Elfving ( 2002).

关键词： projected aggregation methods exact projection incomplete projections oblique projections

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Incomplete oblique projections for solving large inconsistent linear systems

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MATHEMATICAL PROGRAMMING 2008年第1-2期111卷 273-300页

作者： Scolnik, H. D. Echebest, N. Guardarucci, M. T. Vacchino, M. C. Univ Buenos Aires Fac Ciencias Exactas & Nat Dept Computac Buenos Aires DF Argentina Univ Nacl La Plata Fac Ciencias Exactas Dept Matemat La Plata Buenos Aires Argentina

The authors introduced in previously published papers acceleration schemes for projected aggregation methods (PAM), aiming at solving consistent linear systems of equalities and inequalities. They have used the basic idea of forcing each iterate to belong to the aggregate hyperplane generated in the previous iteration. That scheme has been applied to a variety of projection algorithms for solving systems of linear equalities or inequalities, proving that the acceleration technique can be successfully used for consistent problems. The aim of this paper is to extend the applicability of those schemes to the inconsistent case, employing incomplete projections onto the set of solutions of the augmented system Ax - r = b. These extended algorithms converge to the least squares solution. For that purpose, oblique projections are used and, in particular, variable oblique incomplete projections are introduced. They are defined by means of matrices that penalize the norm of the residuals very strongly in the first iterations, decreasing their influence with the iteration counter in order to fulfill the convergence conditions. The theoretical properties of the new algorithms are analyzed, and numerical experiences are presented comparing their performance with several well-known projection methods.

关键词： projected aggregation methods incomplete projections inconsistent system

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Incomplete oblique projections for solving large inconsistent linear systems

Incomplete oblique projections for solving large inconsisten...

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5th Brazilian Workshop on Continuous Optimization

关键词： projected aggregation methods incomplete projections inconsistent system

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