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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)

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)

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)

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)

SUMMARY Algebraic multigrid (AMG) is an iterative method that is often optimal for solving the matrix equations that arise in a wide variety of applications, including discretized partial differential equations. It automatically constructs a sequence of increasingly smaller matrix problems that hopefully enables efficient resolution of all scales present in… (More)

In this paper, we propose new adaptive local refinement (ALR) strategies for first-order system least-squares finite elements in conjunction with algebraic multigrid methods in the context of nested iteration. The goal is to reach a certain error tolerance with the least amount of computational cost and nearly uniform distribution of the error over all… (More)

A significant amount of the computational time in large Monte Carlo simulations of lattice field theory is spent inverting the discrete Dirac operator. Unfortunately, traditional covariant finite difference discretizations of the Dirac operator present serious challenges for standard iterative methods. For interesting physical parameters, the discretized… (More)

Substantial effort has been focused over the last two decades on developing multi-level 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)

Efficient numerical simulation of physical processes is constrained by our ability to solve the resulting linear systems, prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort. Overcoming the limitations of geometric multigrid methods to simple geometries… (More)