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V-cycle Optimal Convergence for DCT-III Matrices

  • C. Tablino Possio
Part of the Operator Theory: Advances and Applications book series (OT, volume 199)

Abstract

The paper analyzes a two-grid and a multigrid method for matrices belonging to the DCT-III algebra and generated by a polynomial symbol. The aim is to prove that the convergence rate of the considered multigrid method (V-cycle) is constant independent of the size of the given matrix. Numerical examples from differential and integral equations are considered to illustrate the claimed convergence properties.

Mathematics Subject Classification (2000)

Primary 65F10 65F15 15A12 

Keywords

DCT-III algebra two-grid and multigrid iterations multi-iterative methods 

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References

  1. [1]
    A. Aricò, M. Donatelli, A V-cycle multigrid for multilevel matrix algebras: proof of optimality. Numer. Math. 105 (2007), no. 4, 511–547 (DOI 10.1007/s00211-006-0049-7).Google Scholar
  2. [2]
    A. Aricò, M. Donatelli, S. Serra-Capizzano, V-cycle optimal convergence for certain (multilevel) structured linear systems. SIAM J. Matrix Anal. Appl. 26 (2004), no. 1, 186–214.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    O. Axelsson, M. Neytcheva, The algebraic multilevel iteration methods — theory and applications. In Proceedings of the Second International Colloquium on Numerical Analysis (Plovdiv, 1993), 13–23, VSP, 1994.Google Scholar
  4. [4]
    R.H. Chan, T.F. Chan, C. Wong, Cosine transform based preconditioners for total variation minimization problems in image processing. In Iterative Methods in Linear Algebra, II, V3, IMACS Series in Computational and Applied Mathematics, Proceedings of the Second IMACS International Symposium on Iterative Methods in Linear Algebra, Bulgaria, 1995, 311–329.Google Scholar
  5. [5]
    R.H. Chan, M. Donatelli, S. Serra-Capizzano, C. Tablino-Possio, Application of multigrid techniques to image restoration problems. In Proceedings of SPIE-Session: Advanced Signal Processing: Algorithms, Architectures, and Implementations XII, Vol. 4791 (2002), F. Luk Ed., 210–221.Google Scholar
  6. [6]
    R.H. Chan, M.K. Ng, Conjugate gradient methods for Toeplitz systems. SIAM Rev. 38 (1996), no. 3, 427–482.Google Scholar
  7. [7]
    R.H. Chan, S. Serra-Capizzano, C. Tablino-Possio, Two-grid methods for banded linear systems from DCT III algebra. Numer. Linear Algebra Appl. 12 (2005), no. 2-3, 241–249.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    G. Fiorentino, S. Serra-Capizzano, Multigrid methods for Toeplitz matrices. Calcolo 28 (1991), no. 3-4, 283–305.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    G. Fiorentino, S. Serra-Capizzano, Multigrid methods for symmetric positive definite block Toeplitz matrices with nonnegative generating functions. SIAM J. Sci. Comput. 17 (1996), no. 5, 1068–1081.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    R. Fischer, T. Huckle, Multigrid methods for anisotropic BTTB systems. Linear Algebra Appl. 417 (2006), no. 2-3, 314–334.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    W. Hackbusch, Multigrid methods and applications. Springer Series in Computational Mathematics, 4. Springer-Verlag, 1985.Google Scholar
  12. [12]
    T. Huckle, J. Staudacher, Multigrid preconditioning and Toeplitz matrices. Electron. Trans. Numer. Anal. 13 (2002), 81–105.zbMATHMathSciNetGoogle Scholar
  13. [13]
    T. Huckle, J. Staudacher, Multigrid methods for block Toeplitz matrices with small size blocks. BIT 46 (2006), no. 1, 61–83.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    X.Q. Jin, Developments and applications of block Toeplitz iterative solvers. Combinatorics and Computer Science, 2. Kluwer Academic Publishers Group, Dordrecht; Science Press, Beijing, 2002.Google Scholar
  15. [15]
    D.G. Luenberger, Introduction to Dynamic Systems: Theory, Models, and Applications, John Wiley & Sons Inc., 1979.Google Scholar
  16. [16]
    M.K. Ng, Iterative methods for Toeplitz systems. Numerical Mathematics and Scientific Computation. Oxford University Press, 2004.Google Scholar
  17. [17]
    M.K. Ng, R.H. Chan, T.F. Chan, A.M. Yip, Cosine transform preconditioners for high resolution image reconstruction. Conference Celebrating the 60th Birthday of Robert J. Plemmons (Winston-Salem, NC, 1999). Linear Algebra Appl. 316 (2000), no. 1-3, 89–104.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    M.K. Ng, R.H. Chan, W.C. Tang, A fast algorithm for deblurring models with Neumann boundary conditions. SIAM J. Sci. Comput. 21 (1999), no. 3, 851–866.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    M.K. Ng, S. Serra-Capizzano, C. Tablino-Possio. Numerical behaviour of multigrid methods for symmetric sinc-Galerkin systems. Numer. Linear Algebra Appl. 12 (2005), no. 2-3, 261–269.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    D. Noutsos, S. Serra-Capizzano, P. Vassalos, Matrix algebra preconditioners for multilevel Toeplitz systems do not insure optimal convergence rate. Theoret. Comput. Sci. 315 (2004), no. 2-3, 557–579.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    J. Ruge, K. Stüben, Algebraic multigrid. In Frontiers in Applied Mathematics: Multigrid Methods. SIAM, 1987, 73–130.Google Scholar
  22. [22]
    S. Serra Capizzano, Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs matrix-sequences. Numer. Math. 92 (2002), no. 3, 433–465.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    S. Serra-Capizzano, Matrix algebra preconditioners for multilevel Toeplitz matrices are not superlinear. Special issue on structured and infinite systems of linear equations. Linear Algebra Appl. 343–344 (2002), 303–319.CrossRefMathSciNetGoogle Scholar
  24. [24]
    S. Serra Capizzano, C. Tablino Possio, Spectral and structural analysis of high precision finite difference matrices for elliptic operators. Linear Algebra Appl. 293 (1999), no. 1-3, 85–131.zbMATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    S. Serra-Capizzano, C. Tablino-Possio, Multigrid methods for multilevel circulant matrices. SIAM J. Sci. Comput. 26 (2004), no. 1, 55–85.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    S. Serra-Capizzano and E. Tyrtyshnikov, How to prove that a preconditioner cannot be superlinear. Math. Comp. 72 (2003), no. 243, 1305–1316.zbMATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    G. Strang, The discrete cosine transform, SIAM Review, 41, n. 1, pp. 135–147, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    H. Sun, X. Jin, Q. Chang, Convergence of the multigrid method for ill-conditioned block Toeplitz systems. BIT 41 (2001), no. 1, 179–190.zbMATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    U. Trottenberg, C.W. Oosterlee and A. Schüller, Multigrid. With contributions by A. Brandt, P. Oswald and K. Stüben. Academic Press, Inc., 2001.Google Scholar
  30. [30]
    E. Tyrtyshnikov, Circulant preconditioners with unbounded inverses. Linear Algebra Appl. 216 (1995), 1–23.zbMATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    R.S. Varga, Matrix Iterative Analysis. Prentice-Hall, Inc., Englewood Cliffs, 1962.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  • C. Tablino Possio
    • 1
  1. 1.Dipartimento di Matematica e ApplicazioniUniversità di Milano BicoccaMilanoItaly

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