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We introduce AMGe, an algebraic multigrid method for solving the discrete equations that arise in Ritz-type nite element methods for partial diierential equations. Assuming access to the element stiiness matrices, AMGe is based on the use of two local measures, which are derived from global measures that appear in existing multigrid theory. These new(More)
Spectral AMGe (ρAMGe), is a new algebraic multigrid method for solving discretizations that arise in Ritz-type finite element methods for partial differential equations. The method assumes access to the element stiffness matrices in order to lessen certain presumptions that can limit other algebraic methods. ρAMGe uses the spectral decomposition of small(More)
The linear systems arising in lattice QCD pose significant challenges for traditional iterative solvers. For physically interesting values of the so-called quark mass, these systems are nearly singular, indicating the need for efficient preconditioners. However, multi-level preconditioners cannot easily be constructed because the Dirac operator associated(More)
Algebraic multigrid (AMG) is currently undergoing a resurgence in popularity, due in part to the dramatic increase in the need to solve physical problems posed on very large, unstructured grids. While AMG has proved its usefulness on various problem types, it is not commonly understood how wide a range of applicability the method has. In this study, we(More)
Substantial effort has been focused over the last two decades on developing multilevel iterative methods capable of solving the large linear systems encountered in engineering practice. These systems often arise from discretizing partial differential equations over unstructured meshes, and the particular parameters or geometry of the physical problem being(More)
Bootstrap Algebraic Multigrid (BAMG) is a multigrid-based solver for matrix equations of the form Ax = b. Its aim is to automatically determine the interpolation weights used in algebraic multigrid (AMG) by locally fitting a set of test vectors that have been relaxed as solutions to the corresponding homogeneous equation, Ax = 0, and are then possibly(More)
This paper develops a multilevel least-squares approach for the numerical solution of the complex scalar exterior Helmholtz equation. This second-order equation is first recast into an equivalent first-order system by introducing several " field " variables. A combination of scaled L 2 and H −1 norms is then applied to the residual of this system to create(More)
A multilevel adaptive aggregation method for calculating the stationary probability vector of an irreducible stochastic matrix is described. The method is a special case of the adap-tive smooth aggregation and adaptive algebraic multigrid methods for sparse linear systems, and is also closely related to certain extensively studied iterative(More)
A smoothed aggregation multigrid method is presented for the numerical calculation of the stationary probability vector of an irreducible sparse Markov chain. It is shown how smoothing the interpolation and restriction operators can dramatically increase the efficiency of aggregation multigrid methods for Markov chains that have been proposed in the(More)