A wavelet-based algebraic multigrid preconditioner for sparse linear systems

This work considers the use of discrete wavelet transform (DWT), based in filters, in the construction of the hierarchy of matrices in the algebraic multigrid method (AMG). The two-dimensional DWT is applied to produce an approximation of the matrix in each level of the wavelets multiresolution decomposition process. In this procedure an operator is created, formed only by lowpass filters, that is applied to the rows and columns of the matrix capturing this approximation. This same operator is used as an intergrid transfer in the AMG. Wavelet-based algebraic multigrid method (WAMG) was implemented, with Daubechies wavelets of orders 6, 4 and 2, and used as a preconditioner for the generalized minimal residual method (GMRES). Numerical results are presented comparing this new approach with the standard algebraic multigrid as preconditioner for sparse linear systems.

[1]  Fábio Henrique Pereira,et al.  A fast algebraic multigrid preconditioned conjugate gradient solver , 2006, Appl. Math. Comput..

[2]  Q. Chang,et al.  On the Algebraic Multigrid Method , 1996 .

[3]  William L. Briggs,et al.  A multigrid tutorial , 1987 .

[4]  Stéphane Mallat,et al.  A Theory for Multiresolution Signal Decomposition: The Wavelet Representation , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[5]  William L. Briggs,et al.  Wavelets and Multigrid , 1993, SIAM J. Sci. Comput..

[6]  Kim Fung Tsang,et al.  Analysis of Electromagnetic Wave Scattering By an Electrically Large Metallic Grating Using Wavelet-Based Algebratic Multigrid Preconditioned Cg Method- Abstract , 2000 .

[7]  Tapan K. Sarkar,et al.  Wavelet applications in engineering electromagnetics , 2002 .

[8]  Albert Cohen,et al.  Wavelet methods in numerical analysis , 2000 .

[9]  Rushan Chen,et al.  Analysis of Millimeter Wave Scattering by an Electrically Large Metallic Grating Using Wavelet-Based Algebraic Multigrid Preconditioned CG Method , 2000 .

[10]  A. Avudainayagam,et al.  Wavelet based multigrid methods for linear and nonlinear elliptic partial differential equations , 2004, Appl. Math. Comput..

[11]  Aria Nosratinia,et al.  Wavelet-Based Image Coding: An Overview , 1999 .

[12]  Trevor J. Terrell.  Introduction to Digital Filters , 1980 .

[13]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[14]  Robert D. Falgout,et al.  Coarse-Grid Selection for Parallel Algebraic Multigrid , 1998, IRREGULAR.