On the Algebraic Multigrid Method

New formulations for the algebraic multigrid (AMG) method are presented. A new interpolation operator is developed, in which the weighting could be negative. Numerical experiments demonstrate that the use of negative interpolation weights is necessary in some applications. New approaches to construct the restriction operator and the coarse-grid equations are discussed. Two new AMG methods are proposed. Theoretical study and convergence analysis of the AMG methods are presented. The main contributions of this paper are to improve the convergence rate and to extend the range of applications of an AMG method. Numerical experiments are reported for matrix computations that resulted from partial differential equations, signal processing, and queueing network problems. The success of the proposed new AMG algorithms is clearly demonstrated by applications to non-diagonally dominant matrix problems for which the standard AMG method fails to converge.