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Annals of Operations Research

, Volume 138, Issue 1, pp 235–257 | Cite as

An Accelerated Iterative Method with Diagonally Scaled Oblique Projections for Solving Linear Feasibility Problems

  • N. Echebest
  • M. T. Guardarucci
  • H. D. Scolnik
  • M. C. Vacchino
Article

Abstract

The Projected Aggregation Methods (PAM) for solving linear systems of equalities and/or inequalities, generate a new iterate xk+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 Garcí a-Palomares and González-Castaño (1998) and also the comparison of its efficiency with that of Censor and Elfving (2002).

Keywords

projected aggregation methods exact projection incomplete projections oblique projections 

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References

  1. Bauschke, H.H. and J.M. Borwein. (1996). “On Projection Algorithms for Solving Convex Feasibility Problems.” SIAM Rev. 38, 367–426.CrossRefGoogle Scholar
  2. Censor, Y. (1988). “Parallel Application of Block-Iterative Methods in Medical Imaging and Radiation Therapy.” Math. Programming 42, 307–325.CrossRefGoogle Scholar
  3. Censor, Y., D. Gordon, and R. Gordon. (2001a). “Component Averaging: An Efficient Iterative Parallel Algorithm for Large and Sparse Unstructured Problems.” Parallel Computing 27, 777–808.CrossRefGoogle Scholar
  4. Censor, Y., D. Gordon, and R. Gordon. (2001b). “BICAV: An Inherently Parallel Algorithm for Sparse Systems with Pixel-Dependent Weighting.” IEEE Trans. on Medical Imaging 20, 1050–1060.CrossRefGoogle Scholar
  5. Censor, Y. and T. Elfving. (2002). “Block-Iterative Algorithms with Diagonally Scaled Oblique Projections for the Linear Feasibility Problem.” SIAM Journal on Matrix Analysis and Applications 24, 40– 58.Google Scholar
  6. Echebest, N., M.T. Guardarucci, H.D. Scolnik, and M.C. Vacchino. (2004). “An Acceleration Scheme for Solving Convex Feasibility Problems Using Incomplete Projection Algorithms.” Numerical Algorithms 35, 335–350.CrossRefGoogle Scholar
  7. Garcí a-Palomares, U.M. (1993). “Parallel Projected Aggregation Methods for Solving the Convex Feasibility Problem.” SIAM J. Optim. 3, 882–900.CrossRefGoogle Scholar
  8. Garcí a-Palomares, U.M. and F.J. González-Castaño. (1998). “Incomplete Projection Algorithms for Solving the Convex Feasibility Problem.” Numerical Algorithms 18, 177–193.CrossRefGoogle Scholar
  9. Gubin, L.G., B.T. Polyak, and E.V. Raik. (1967). “The Method of Projections for Finding the Common Point of Convex Sets.” USSR Comput. Math. and Math.Phys. 7, 1–24.CrossRefGoogle Scholar
  10. Iusem, A.N. and A. De Pierro. (1986). “Convergence Results for an Accelerated Nonlinear Cimmino Algorithm.” Numer. Math. 49, 367–378.CrossRefGoogle Scholar
  11. Herman, G.T. and L.B. Meyer. (1993). “Algebraic Reconstruction Techniques can be Made Computationally Efficient.” IEEE Trans. Medical Imaging 12, 600–609.CrossRefGoogle Scholar
  12. Saad, Y. (l990). “SPARSKIT: A Basic Tool Kit for Sparse Matrix Computations.” Technical Report 90-20, Research Institute for Avanced Computer Science. NASA Ames Research Center, Moffet Field, CA.Google Scholar
  13. Scolnik, H., N. Echebest, M.T. Guardarucci, and M.C. Vacchino. (2001). “New Optimized and Accelerated PAM Methods for Solving Large Non-symmetric Linear Systems: Theory and Practice.” In D. Butnariu, Y. Censor, and S. Reich (eds.), Inherently Parallel Algorithms in Feasibility and Optimization and their Applications, Studies in Computational Mathematics. Amsterdam: Elsevier Science, Volume 8, pp. 457–470.Google Scholar
  14. Scolnik, H., N. Echebest, M.T. Guardarucci, and M.C. Vacchino. (2002a). “A Class of Optimized Row Projection Methods for Solving Large Non-Symmetric Linear Systems.” Applied Numerical Mathematics 41, 499–513.CrossRefGoogle Scholar
  15. Scolnik, H., N. Echebest, M.T. Guardarucci, and M.C. Vacchino. (2002b). “Acceleration Scheme for Parallel Projected Aggregation Methods for Solving Large Linear Systems.” Annals of Operations Research 117, 95–115.CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • N. Echebest
    • 1
  • M. T. Guardarucci
    • 1
  • H. D. Scolnik
    • 2
  • M. C. Vacchino
    • 2
  1. 1.Departamento de MatemáaticaFacultad de Ciencias Exactas, Universidad Nacional de La PlataArgentina
  2. 2.Departamento de ComputaciónFacultad de Ciencias, Exactas y Naturales, Universidad de Buenos AiresArgentina

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