A linear acceleration row action method for projecting onto subspaces

Electronic Transactions on Numerical Analysis

Volumn 20, Issue , 2005, Pages 253-275

  Appleby, Glenn ^a   Smolarski, Dennis C ^a  

^a SANTA CLARA UNIVERSITY   (United States)

Author keywords

Cimmino; Dax; Iterative methods; Kaczmarz; Linear systems; Pierra; Projection methods; Row action methods

Indexed keywords

EID: 29344442838     PISSN: 10689613     EISSN: 10689613     Source Type: Journal    
DOI: None     Document Type: Article

References (39)

1. R. ANSORGE, Connections between the Cimmino-method and the Kaczmarz-method for the solution of singular and regular systems of equations, Computing, 33 (1984), pp. 367-375.
2. M. ARIOLI, I. DUFF, J. NOAILLES, AND D. RUIZ, A block projection method for sparse matrices, STAM J. Sci. Statis. Comput., 13 (1992), pp. 47-70.
3. S. F. ASHBY, T. A. MANTEUFFEL, AND P. E. SAYLOR, A taxonomy for conjugate gradient methods, SIAM J. Numer. Anal., 27 (1990), pp. 1542-1568.
4. H. H. BAUSCHKE AND J. M. BORWEIN, On projection algorithms for solving convex feasibility problems, SIAM Rev., 38 (1996), pp. 367-426.
5. M. BENZI, G. H. GOLUB, AND J. LIESEN, Numerical solution of saddle point problems, Acta Numer., (2005), pp. 1-137.
6. A. BJÖRCK AND T. ELFVING, Accelerated projection methods for comuting pseudoinverse solutions of systems of linear equations, BIT, 19 (1979), pp. 145-163.
7. E. BODEWIG, Matrix Calculus, North-Holland Publ. Co., Amsterdam, 1959.
8. R. BRAMLEY, An orthogonal projection algorithm for generalized Stokes problems, Center for Supercomputing Research and Development, University of Illinois at Urbana-Champaign, January 1992, Report No. 1190.
9. R. BRAMLEY AND A. S AMEH, Row projection methods for large nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 13 (1992), pp. 168-193.
10. Y. CENSOR, Row-action methods for huge and sparse systems and their applications, SIAM Rev., 23 (1981), pp. 444-466.
11. _, Finite series-expansion reconstruction methods, Proceedings of the IEEE, 71 (1983), pp. 409-419.
12. _, Parallel application of block-iterative methods in medical imaging and radiation therapy, Math. Programming, 42 (1988), pp. 307-25.
13. Y. CENSOR, M. D. ALTSCHULER, AND W. D. POWLIS, On the use of Cimmino's simultaneous projections method for computing a solution of the inverse problem in radiation therapy treatment planning, Inverse Problems, 4 (1988), pp. 607-23.
14. Y. CENSOR, P. P. B. EGGERMONT, AND D. GORDON, Strong underrelaxation in Kaczmarz's method for inconsistent systems, Numer. Math., 41 (1983), pp. 83-92.
15. Y. CENSOR, D. GORDON, AND R. GORDON, Component averaging: An efficient iterative parallel algorithm for large and sparse unstructured problems, Paral. Comput., 27 (2001), pp. 777-808.
16. Y. CENSOR AND G. T. HERMAN, On some optimization techniques in image reconstruction from projections, Appl. Numer. Math., 3 (1987), pp. 365-91.
17. Y. CENSOR AND S. A. ZENIOS, Parallel Optimization: Theory, Algorithms and Applications, Oxford University Press, New York, 1997.
18. G. CIMMlNO, Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari, Ric. Sci. progr. tecn. econom. naz., XVI (1938), pp. 326-333.
19. A. DAX, The convergence of linear stationary Iterative methods for solving singular unstructured systems of linear equations, SIAM Rev., 32 (1990), pp. 611-635.
20. A. DAX, Line search accleration of iterative methods, Lin. Alg. Appl., 130 (1990), pp. 43-63.
21. H. C. ELMAN AND G. H. GOLUB, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Sci. Stat. Comput., 31 (1994), pp. 1645-1661.
22. G. H. GOLUB, C. F. VANLOAN, Matrix Computations (Third ed.), Baltimore: The Johns Hopkins University Press, 1996.
23. R. GORDON, R. BENDER, AND G. T. HERMAN, Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography, J. Theoret. Biol., 29 (1970), pp. 471-481.
24. A. S. HOUSEHOLDER, The Theory of Matrices in Numerical Analysis, New York: Dover Publications, 1975.
25. M. JIANG AND G. WANG, Development of iterative algorithms for image reconstruction, J. X-ray Sci. Tech., 10 (2001), pp. 77-86.
26. M. JIANO AND G. WANG, Convergence studies on iterative algorithms for image reconstruction,IEEE Trans. Med. Imaging, 22 (2003), pp. 569-579.
27. S. KACZMARZ, Angenäherte Auflösung von Systemen linearer Gleichungen, Bull. Intern. Acad. Polon. Sci. Lett., Ser. A, 35 (1937), pp. 355-357.
28. A. C. KAK AND M. SLANEY, Principles of Computerized Tomographic Imaging, IEEE Press, New York, 1988.
29. C. KAMATH AND S. WEERATUNGA, Projection methods for the numerical solution of non-self-adjoint elliptic partial differential eauations, Num. Methods for P.D.E., 8 (1992), pp. 59-76.
30. S. L. LEE, Krylov methods for the numerical solution of initial-value problems in differential-algebraic equations, Department of Computer Science, University of Illinois at Urbana-Champaign, December 1993, Report No. UIUCDCS-R-93-1814.
31. K. MULLER, R. YAGEL AND J. J. WHELLER, Anti-aliased 3D cone-beam reconstruction of low-contrast objects with algebraic methods, IEEE Trans. Med. Imaging, 18 (1999), pp. 519-537.
32. F. NATTERER, The Mathematics of Computerized Tomography, New York: John Wiley and Sons, 1986. Reprinted: Philadelphia: SIAM, 2001.
33. G. PIERRA, Decomposition through formalization in a product space, Math. Prog., 28 (1984), pp. 96-118.
34. C. POPA, Preconditioned Kaczmarz-extended algorithm with relaxation parameters, Korean J. Comp. Appl. Math., 6 (1999), pp. 523-535.
35. Y. SAAD AND H. A. VAN DER VORST, Iterative solution of linear systems in the 20th century, J. Comp. Appl. Math., 123 (2000), pp. 1-33.
36. K. TANABE, Projection method for solving a singular system of linear equations and its applications, Numer. Math., 17 (1971), pp. 203-214.
37. L. N. TREFETHEN AND D. BAU, III, Numerical Linear Algebra, Philadelphia: SIAM, 1997.
38. M. R. TRUMMER, Reconstructing pictures from projections: On the convergence of the ART algorithm with relaxation, Computing, 26 (1981), pp. 189-195.
39. M. R. TRUMMER, A note on the ART of relaxation, Computing, 33 (1984), pp. 349-352.