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Abstract:

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 Garcí a-Palomares and González-Castaño (1998) and also the comparison of its efficiency with that of Censor and Elfving (2002). © 2005 Springer Science + Business Media, Inc.

Registro:

Documento: Artículo
Título:An accelerated iterative method with diagonally scaled oblique projections for solving linear feasibility problems
Autor:Echebest, N.; Guardarucci, M.T.; Scolnik, H.D.; Vacchino, M.C.
Filiación:Departamento de Matemáatica, Facultad de Ciencias Exactas, Universidad Nacional de la Plata, Argentina
Departamento de Computación, Facultad de Ciencias, Exactas y Naturales, Universidad de Buenos Aires, Argentina
Palabras clave:Exact projection; Incomplete projections; Oblique projections; Projected aggregation methods
Año:2005
Volumen:138
Número:1
Página de inicio:235
Página de fin:257
DOI: http://dx.doi.org/10.1007/s10479-005-2456-z
Título revista:Annals of Operations Research
Título revista abreviado:Ann. Oper. Res.
ISSN:02545330
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_02545330_v138_n1_p235_Echebest

Referencias:

  • Bauschke, H.H., Borwein, J.M., On projection algorithms for solving convex feasibility problems (1996) SIAM Rev., 38, pp. 367-426
  • Censor, Y., Parallel application of block-iterative methods in medical imaging and radiation therapy (1988) Math. Programming, 42, pp. 307-325
  • Censor, Y., Gordon, D., Gordon, R., Component averaging: An efficient iterative parallel algorithm for large and sparse unstructured problems (2001) Parallel Computing, 27, pp. 777-808
  • Censor, Y., Gordon, D., Gordon, R., BICAV: An inherently parallel algorithm for sparse systems with pixel-dependent weighting (2001) IEEE Trans. on Medical Imaging, 20, pp. 1050-1060
  • Censor, Y., Elfving, T., Block-iterative algorithms with diagonally scaled oblique projections for the linear feasibility problem (2002) SIAM Journal on Matrix Analysis and Applications, 24, pp. 40-58
  • Behebest, N., Guardarucci, M.T., Scolnik, H.D., Vacchino, M.C., An acceleration scheme for solving convex feasibility problems using incomplete projection algorithms (2004) Numerical Algorithms, 35, pp. 335-350
  • García-Palomares, U.M., Parallel projected aggregation methods for solving the convex feasibility problem (1993) SIAM J. Optim., 3, pp. 882-900
  • García-Palomares, U.M., González-Castaño, F.J., Incomplete projection algorithms for solving the convex feasibility problem (1998) Numerical Algorithms, 18, pp. 177-193
  • Gubin, L.G., Polyak, B.T., Raik, E.V., The method of projections for finding the common point of convex sets (1967) USSR Comput. Math. and Math.Phys., 7, pp. 1-24
  • Iusem, A.N., De Pierro, A., Convergence results for an accelerated nonlinear cimmino algorithm (1986) Numer. Math., 49, pp. 367-378
  • Herman, G.T., Meyer, L.B., Algebraic reconstruction techniques can be made computationally efficient (1993) IEEE Trans. Medical Imaging, 12, pp. 600-609
  • Saad, Y., SPARSKIT: A basic tool kit for sparse matrix computations (1990) Technical Report, 90 (20). , Research Institute for Avanced Computer Science. NASA Ames Research Center, Moffet Field, CA
  • Scolnik, H., Behebest, N., Guardarucci, M.T., Vacchino, M.C., New optimized and accelerated PAM methods for solving large non-symmetric linear systems: Theory and practice (2001) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, 8, pp. 457-470. , D. Butnariu, Y. Censor, and S. Reich (eds.), Studies in Computational Mathematics. Amsterdam: Elsevier Science
  • Scolnik, H., Echebest, N., Guardarucci, M.T., Vacchino, M.C., A class of optimized row projection methods for solving large non-symmetric linear systems (2002) Applied Numerical Mathematics, 41, pp. 499-513
  • Scolnik, H., Behebest, N., Guardarucci, M.T., Vacchino, M.C., Acceleration scheme for parallel projected aggregation methods for solving large linear systems (2002) Annals of Operations Research, 117, pp. 95-115

Citas:

---------- APA ----------
Echebest, N., Guardarucci, M.T., Scolnik, H.D. & Vacchino, M.C. (2005) . An accelerated iterative method with diagonally scaled oblique projections for solving linear feasibility problems. Annals of Operations Research, 138(1), 235-257.
http://dx.doi.org/10.1007/s10479-005-2456-z
---------- CHICAGO ----------
Echebest, N., Guardarucci, M.T., Scolnik, H.D., Vacchino, M.C. "An accelerated iterative method with diagonally scaled oblique projections for solving linear feasibility problems" . Annals of Operations Research 138, no. 1 (2005) : 235-257.
http://dx.doi.org/10.1007/s10479-005-2456-z
---------- MLA ----------
Echebest, N., Guardarucci, M.T., Scolnik, H.D., Vacchino, M.C. "An accelerated iterative method with diagonally scaled oblique projections for solving linear feasibility problems" . Annals of Operations Research, vol. 138, no. 1, 2005, pp. 235-257.
http://dx.doi.org/10.1007/s10479-005-2456-z
---------- VANCOUVER ----------
Echebest, N., Guardarucci, M.T., Scolnik, H.D., Vacchino, M.C. An accelerated iterative method with diagonally scaled oblique projections for solving linear feasibility problems. Ann. Oper. Res. 2005;138(1):235-257.
http://dx.doi.org/10.1007/s10479-005-2456-z