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.