A V-cycle Multigrid for multilevel matrix algebras: Proof of optimality

Numerische Mathematik

Volumn 105, Issue 4, 2007, Pages 511-547

  Aricò, Antonio ^a   Donatelli, Marco ^a  

^a UNIVERSITY OF INSUBRIA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33846798182     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00211-006-0049-7     Document Type: Article

References (31)

1. Aricò, A., Donatelli, M., Serra Capizzano, S.: V-cycle optimal convergence for certain (multi-level) structured linear system. SIAM J. Matrix Anal. Appl. 26(1), 186-214 (2004)
2. Axelsson, G., Neytcheva, M.: The algebraic multilevel iteration methods-theory and applications. In: Bainov, D. (ed.) Proceedings of the second int. Coll. on numerical analysis, VSP 1994, Bulgaria (1993)
3. Beckermann, B., Serra Capizzano, S.: On the asymptotic spectrum of Finite Elements matrices. SIAM J. Numer. Anal. (to appear)
4. Bini, D., Capovani, M.: Spectral and computational properties of band simmetric Toeplitz matrices. Linear Algebra Appl. 52/53, 99-125 (1983)
5. Bini, D., Favati, P.: On a matrix algebra related to the discrete Hartley transform. SIAM J. Matrix Anal. Appl. 14(2), 500-507 (1993)
6. Bramble, J.H.: Multigrid methods. volume 294 of Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Harlow, (1993)
7. Bramble, J.H., Pasciak, J.E., Wang, J., Xu, J.: Convergence estimates for product iterative methods with applications to domain decomposition. Math. Comput. 57(195), 1-21 (1991)
8. Bramble, J.H., Pasciak, J.E., Wang, J., Xu, J.: Convergence estimates for multigrid algorithms without regularity assumptions. Math. Comput. 57(195) 23-45 (1991)
9. Brandt, A.: Rigorous quantitative analysis of multigrid, I constant coefficients two-level cycle with L2-norm. SIAM J. Numer. Anal. 31(6), 1695-1730 (1994)
10. Chan, R.H., Chang, Q., Sun, H.: Multigrid method for ill-conditioned symmetric Toeplitz systems. SIAM J. Sci. Comput. 19(2), 516-529 (1998)
11. Chan, R.H., Ng, M.: Conjugate gradient methods for Toeplitz systems. SIAM Rev. 38, 427-482 (1996)
12. Chan, R.H., Serra Capizzano, S., Tablino Possio, C.: Two-Grid methods for banded linear systems from DCT III algebra. Numer. Linear Algebra Appl. 12(2/3), 241-249 (2005)
13. Chang, Q., Jin, X., Sun, H.: Convergence of the multigrid method for ill-conditioned block toeplitz systems. BIT 41(1), 179-190 (2001)
14. Donatelli, M.: A Multigrid method for image restoration with Tikhonov regularization. Numer. Linear Algebra Appl. 12, 715-729 (2005)
15. Donatelli, M., Serra Capizzano, S.: On the regularizing power of multigrid-type algorithms. SIAM J. Sci. Comput. 26(6), 2053-2076 (2006)
16. Fiorentino, G., Serra Capizzano, S.: Multigrid methods for Toeplitz matrices. Calcolo 28, 283-305 (1991)
17. Fiorentino, G., Serra Capizzano, S.: Multigrid methods for symmetric positive definite block Toeplitz matrices with nonnegative generating functions. SIAM J. Sci. Comp. 17(4), 1068-1081 (1996)
18. Heinz, E., Hanke, M., Neubauer, A.: Regularization of Inverse Problems, Mathematics and its Applications. Kluwer, Dordrecht (1996)
19. Huckle, T., Staudacher, J.: Multigrid preconditioning and Toeplitz matrices. Electr. Trans. Numer. Anal. 13, 81-105 (2002)
20. Kalouptsidis, N., Carayannis, G., Manolakis, D.: Fast algorithms for block Toeplitz matrices with Toeplitz entries. Signal Process. 6, 77-81 (1984)
21. Ruge, J.W., Stüben, K.: Algebraic multigrid. In: McCormick, S. (ed.) Frontiers in Applied Mathemathics: Multigrid Methods. SIAM, Philadelphia (1987), pp. 73-130
22. Serra Capizzano, S.: Preconditioning strategies for asymptotically ill-conditioned block Toeplitz systems. BIT 34(4), 579-594 (1994)
23. Serra Capizzano, S.: Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs Matrix-sequences. Numer. Math. 92(3), 433-465 (2002)
24. Serra Capizzano, S.: A note on anti-reflective boundary conditions and fast deblurring models. SIAM J. Sci. Comput. 25(4), 1307-1325 (2004)
25. Serra Capizzano, S., Tablino Possio, C.: Multigrid methods for multilevel circulant matrices. SIAM J. Sci. Comp. 26(1), 55-85 (2005)
26. Serra Capizzano, S., Tyrtyshnikov, E.: Any circulant-like preconditioner for multilevel matrices is not superlinear. SIAM J. Matrix Anal. Appl. 21(2), 431-439 (1999)
27. Trottenberg, U., Oosterlee, C.W., Schüller, A.: Multigrid. Academic, New York (2001)
28. Tyrtyshnikov, E.: Circulant precondictioners with unbounded inverse. Linear Algebra Appl. 216, 1-23 (1995)
29. Tyrtyshnikov, E.: A unifying approach to some old and new theorems on preconditioning and clustering. Linear Algebra Appl. 232, 1-43 (1996)
30. Varga, R.S.: Matrix Iterative Analysis. Prentice Hall, Englewood Cliffs (1962)
31. Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34(4), 581-613 (1992)