ANALYSIS OF SMOOTHED AGGREGATION MULTIGRID METHODS BASED ON TOEPLITZ MATRICES ∗

Aim of the paper is to analyze multigrid methods based on smoothed aggregation in the case of circulant and Toeplitz matrices. The analysis is based on the classical convergence theory for these types of matrices and results in optimal smoothing parameters that have to be chosen for the smoothing of the grid transfer operators in order to guarantee optimality of the resulting multigrid method. The developed analysis allows a new understanding of smoothed aggregation and can also be applied for unstructured matrices. A detailed analysis of the multigrid convergence behavior is developed for the finite difference discretization of the 2D Laplacian with nine point stencils. The theoretical findings are backed up by numerical experiments.

[1]  Stefano Serra Capizzano,et al.  Multigrid Methods for Symmetric Positive Definite Block Toeplitz Matrices with Nonnegative Generating Functions , 1996, SIAM J. Sci. Comput..

[2]  Marco Donatelli,et al.  Generalized grid transfer operators for multigrid methods applied on Toeplitz matrices , 2015 .

[3]  T. Manteuffel,et al.  Adaptive Smoothed Aggregation ( α SA ) Multigrid ∗ , 2005 .

[4]  Thomas Huckle,et al.  Multigrid Preconditioning and Toeplitz Matrices , 2002 .

[5]  J. Mandel,et al.  A local convergence proof for the iterative aggregation method , 1983 .

[6]  Stefano Serra Capizzano,et al.  Preliminary Remarks on Multigrid Methods for Circulant Matrices , 2000, NAA.

[7]  Marian Brezina,et al.  Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems , 2005, Computing.

[8]  S. F. McCormick,et al.  Multigrid Methods for Variational Problems , 1982 .

[9]  D. Bartuschat.  Algebraic Multigrid , 2007 .

[10]  Stefano Serra Capizzano,et al.  V-cycle Optimal Convergence for Certain (Multilevel) Structured Linear Systems , 2004, SIAM J. Matrix Anal. Appl..

[11]  Marco Donatelli,et al.  A V-cycle Multigrid for multilevel matrix algebras: proof of optimality , 2007, Numerische Mathematik.

[12]  Thomas Huckle,et al.  Multigrid Methods for Block Toeplitz Matrices with Small Size Blocks , 2006 .

[13]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[14]  Stefano Serra Capizzano,et al.  Multigrid Methods for Multilevel Circulant Matrices , 2004, SIAM J. Sci. Comput..

[15]  Artem Napov,et al.  Algebraic analysis of aggregation‐based multigrid , 2011, Numer. Linear Algebra Appl..

[16]  Yvan Notay,et al.  Analysis of Aggregation-Based Multigrid , 2008, SIAM J. Sci. Comput..

[17]  Stephen F. McCormick,et al.  Multigrid Methods for Variational Problems: General Theory for the V-Cycle , 1985 .

[18]  Justin W. L. Wan,et al.  Practical Fourier analysis for multigrid methods , 2007, Math. Comput..

[19]  Jan Mandel,et al.  An algebraic theory for multigrid methods for variational problems , 1988 .

[20]  Marco Donatelli,et al.  Multigrid methods for Toeplitz linear systems with different size reduction , 2010, 1010.5730.

[21]  Panayot S. Vassilevski,et al.  An improved convergence analysis of smoothed aggregation algebraic multigrid , 2012, Numer. Linear Algebra Appl..

[22]  Stefano Serra Capizzano,et al.  Numerische Mathematik Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs Matrix-sequences , 2002 .

[23]  Dietrich Braess.  Towards algebraic multigrid for elliptic problems of second order , 2005, Computing.

[24]  Marco Donatelli,et al.  An algebraic generalization of local Fourier analysis for grid transfer operators in multigrid based on Toeplitz matrices , 2010, Numer. Linear Algebra Appl..

[25]  W. Miranker,et al.  Acceleration by aggregation of successive approximation methods , 1982 .

[26]  Thomas Huckle,et al.  Multigrid methods for anisotropic BTTB systems , 2006 .

[27]  Stefano Serra,et al.  Multigrid methods for toeplitz matrices , 1991 .