Abstract
We analyze the convergence rate of a multigrid method for multilevel linear systems whose coefficient matrices are generated by a real and nonnegative multivariate polynomial f and belong to multilevel matrix algebras like circulant, tau, Hartley, or are of Toeplitz type. In the case of matrix algebra linear systems, we prove that the convergence rate is independent of the system dimension even in presence of asymptotical ill-conditioning (this happens iff f takes the zero value). More precisely, if the d-level coefficient matrix has partial dimension n r at level r, with \({r=1, \ldots, d}\) , then the size of the system is \({{N(\varvec{n})=\prod_{r=1}^d n_r}}\) , \({\varvec{n}=(n_1, \ldots, n_d)}\) , and O(N(n)) operations are required by the considered V-cycle Multigrid in order to compute the solution within a fixed accuracy. Since the total arithmetic cost is asymptotically equivalent to the one of a matrix-vector product, the proposed method is optimal. Some numerical experiments concerning linear systems arising in 2D and 3D applications are considered and discussed.
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Aricò A., Donatelli M., Serra Capizzano S. (2004) V-cycle optimal convergence for certain (multilevel) structured linear system. SIAM J. Matrix Anal. Appl. 26(1): 186–214
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)
Beckermann, B., Serra Capizzano, S.: On the asymptotic spectrum of Finite Elements matrices. SIAM J. Numer. Anal. (to appear)
Bini D., Capovani M. (1983) Spectral and computational properties of band simmetric Toeplitz matrices. Linear Algebra Appl. 52/53: 99–125
Bini D., Favati P. (1993) On a matrix algebra related to the discrete Hartley transform. SIAM J. Matrix Anal. Appl. 14(2): 500–507
Bramble, J.H.: Multigrid methods. volume 294 of Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Harlow, (1993)
Bramble J.H., Pasciak J.E., Wang J., Xu J. (1991) Convergence estimates for product iterative methods with applications to domain decomposition. Math. Comput. 57(195): 1–21
Bramble J.H., Pasciak J.E., Wang J., Xu J. (1991) Convergence estimates for multigrid algorithms without regularity assumptions. Math. Comput. 57(195): 23–45
Brandt A. (1994) Rigorous quantitative analysis of multigrid, I constant coefficients two-level cycle with L2-norm. SIAM J. Numer. Anal. 31(6): 1695–1730
Chan R.H., Chang Q., Sun H. (1998) Multigrid method for ill-conditioned symmetric Toeplitz systems. SIAM J. Sci. Comput. 19(2): 516–529
Chan R.H., Ng M. (1996) Conjugate gradient methods for Toeplitz systems. SIAM Rev. 38: 427–482
Chan R.H., Serra Capizzano S., Tablino Possio C. (2005) Two-Grid methods for banded linear systems from DCT III algebra. Numer. Linear Algebra Appl. 12(2/3): 241–249
Chang Q., Jin X., Sun H. (2001) Convergence of the multigrid method for ill-conditioned block toeplitz systems. BIT 41(1): 179–190
Donatelli M. (2005) A Multigrid method for image restoration with Tikhonov regularization. Numer. Linear Algebra Appl. 12: 715–729
Donatelli M., Serra Capizzano S. (2006) On the regularizing power of multigrid-type algorithms. SIAM J. Sci. Comput. 26(6): 2053–2076
Fiorentino G., Serra Capizzano S. (1991) Multigrid methods for Toeplitz matrices. Calcolo 28: 283–305
Fiorentino G., Serra Capizzano S. (1996) Multigrid methods for symmetric positive definite block Toeplitz matrices with nonnegative generating functions. SIAM J. Sci. Comp. 17(4): 1068–1081
Heinz E., Hanke M., Neubauer A. (1996) Regularization of Inverse Problems, Mathematics and its Applications. Kluwer, Dordrecht
Huckle T., Staudacher J. (2002) Multigrid preconditioning and Toeplitz matrices. Electr. Trans. Numer. Anal. 13: 81–105
Kalouptsidis N., Carayannis G., Manolakis D. (1984) Fast algorithms for block Toeplitz matrices with Toeplitz entries. Signal Process. 6: 77–81
Ruge J.W., Stüben K. (1987) Algebraic multigrid. In: McCormick S. (eds) Frontiers in Applied Mathemathics: Multigrid Methods. SIAM, Philadelphia, pp. 73–130
Serra Capizzano S. (1994) Preconditioning strategies for asymptotically ill-conditioned block Toeplitz systems. BIT 34(4): 579–594
Serra Capizzano S. (2002) Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs Matrix-sequences. Numer. Math. 92(3): 433–465
Serra Capizzano S. (2004) A note on anti-reflective boundary conditions and fast deblurring models. SIAM J. Sci. Comput. 25(4): 1307–1325
Serra Capizzano S., Tablino Possio C. (2005) Multigrid methods for multilevel circulant matrices. SIAM J. Sci. Comp. 26(1): 55–85
Serra Capizzano S., Tyrtyshnikov E. (1999) Any circulant-like preconditioner for multilevel matrices is not superlinear. SIAM J. Matrix Anal. Appl. 21(2): 431–439
Trottenberg U., Oosterlee C.W., Schüller A. (2001) Multigrid. Academic, New York
Tyrtyshnikov E. (1995) Circulant precondictioners with unbounded inverse. Linear Algebra Appl. 216: 1–23
Tyrtyshnikov E. (1996) A unifying approach to some old and new theorems on precondictioning and clustering. Linear Algebra Appl. 232: 1–43
Varga R.S. (1962) Matrix Iterative Analysis. Prentice Hall, Englewood Cliffs
Xu J. (1992) Iterative methods by space decomposition and subspace correction. SIAM Rev. 34(4): 581–613
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Aricò, A., Donatelli, M. A V-cycle Multigrid for multilevel matrix algebras: proof of optimality. Numer. Math. 105, 511–547 (2007). https://doi.org/10.1007/s00211-006-0049-7
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DOI: https://doi.org/10.1007/s00211-006-0049-7