Steve F. McCormick

Adaptive Algebraic Multigrid

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 and differential equations, algebraic multigrid methods construct the multigrid hierarchy based only on the given matrix. While this allows for efficient black-box solution of the linear systems associated with discretizations of many elliptic differential equations, it also results in a lack of robustness due to unsatisfied assumptions made on the near null spaces of these matrices. This paper introduces an extension to algebraic multigrid methods that removes the need to make such assumptions by utilizing an adaptive process. Emphasis is on the principles that guide the adaptivity and their application to algebraic multigrid solution of certain symmetric positive-definite linear systems.

Smoothed Aggregation Multigrid for Markov Chains

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 literature. The proposed smoothing approach is inspired by smoothed aggregation multigrid for linear systems, supplemented with a new lumping technique that assures well-posedness of the coarse-level problems: the coarse-level operators are singular M-matrices on all levels, resulting in strictly positive coarse-level corrections on all levels. Numerical results show how these methods lead to nearly optimal multigrid efficiency for an extensive set of test problems, both when geometric and algebraic aggregation strategies are used.

Adaptive reduction-based AMG

SUMMARY With the ubiquity of large-scale computing resources has come significant attention to practical details of fast algorithms for the numerical solution of partial dierential equations. Included in this group are the class of multigrid and algebraic multigrid algorithms that are eective solvers for many of the large matrix problems arising from the discretization of elliptic operators. Algebraic multigrid (AMG) is especially eective for many problems with discontinuous coecients, discretized on unstructured grids, or over complex geometries. While much eort has been invested in improving the practical performance of AMG, little theoretical understanding of this performance has emerged. This paper presents a two-level convergence theory for a reduction-based variant of AMG, called AMGr, which is particularly appropriate for linear systems that have M-matrix-like properties. For situations where less is known about the problem matrix, an adaptive version of AMGr that automatically determines the form of the reduction needed by the AMGr process is proposed. The adaptive cycle is shown, in both theory and practice, to yield an eective AMGr algorithm. Copyright c 2006 John Wiley & Sons, Ltd.

Algebraic Multigrid for Markov Chains

An algebraic multigrid (AMG) method is presented for the calculation of the stationary probability vector of an irreducible Markov chain. The method is based on standard AMG for nonsingular linear systems, but in a multiplicative, adaptive setting. A modified AMG interpolation formula is proposed that produces a nonnegative interpolation operator with unit row sums. We show how the adoption of a previously described lumping technique maintains the irreducible singular M-matrix character of the coarse-level operators on all levels. Together, these properties are sufficient to guarantee the well-posedness of the algorithm. Numerical results show how it leads to nearly optimal multigrid efficiency for a representative set of test problems.

Spectral Element Agglomerate AMGe

The purpose of this note is to describe an algorithm resulting from the uniting of two ideas introduced and applied elsewhere. For many problems, AMG has always been difficult due to complexities whose natures are difficult to discern from the entries of matrix A alone. Element-based interpolation has been shown to be an effective method for some of these problems, but it requires access to the element matrices on all levels. One way to obtain these has been to perform element agglomeration to form coarse elements, but in complicated situations defining the coarse degrees of freedom (dofs) is not easy. The spectral approach to coarse dof selection is very attractive due to its elegance and simplicity. The algorithm presented here combines the robustness of element interpolation, the ease of coarsening by element agglomeration, and the simplicity of defining coarse dofs through the spectral approach. As demonstrated in the numerical results, the method does yield a reasonable solver for the problems described. It can, however, be an expensive method due to the number and cost of the local, small dense linear algebra problems; making it a generally competitive method remains an area for further research.

A robust multilevel approach for minimizing H(div)-dominated functionals in an H1-conforming finite element space

The standard multigrid algorithm is widely known to yield optimal convergence whenever all high-frequency error components correspond to large relative eigenvalues. This property guarantees that smoothers like Gauss–Seidel and Jacobi will significantly dampen all the high-frequency error components, and thus, produce a smooth error. This has been established for matrices generated from standard discretizations of most elliptic equations. In this paper, we address a system of equations that is generated from a perturbation of the non-elliptic operator I-grad div by a negative e Δ. For enear to one, this operator is elliptic, but as eapproaches zero, the operator becomes non-elliptic as it is dominated by its non-elliptic part. Previous research on the non-elliptic part has revealed that discretizing I-grad div with the proper finite element space allows one to define a robust geometric multigrid algorithm. The robustness of the multigrid algorithm depends on a relaxation operator that yields a smooth error. We use this research to assist in developing a robust discretization and solution method for the perturbed problem. To this end, we introduce a new finite element space for tensor product meshes that is used in the discretization, and a relaxation operator that succeeds in dampening all high-frequency error components. The success of the corresponding multigrid algorithm is first demonstrated by numerical results that quantitatively imply convergence for any eis bounded by the convergence for eequal to zero. Then we prove that convergence of this multigrid algorithm for the case of e equal to zero is independent of mesh size. Copyright

Operator‐based interpolation for bootstrap algebraic multigrid

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 improved later using a multilevel eigensolver. This paper introduces a flexible variant of BAMG that determines the interpolation weights indirectly by ‘collapsing’ the unwanted connections in ‘operator interpolation’. Compared to BAMG, this indirect BAMG approach (iBAMG) is more in the spirit of classical AMG, which collapses unwanted connections in operator interpolation based on the (restrictive) assumption that smooth error is locally constant. This paper studies the numerical performance of iBAMG and establishes an equivalence, under certain assumptions, between it and a slightly modified (standard) BAMG scheme. To focus on camparing BAMG and iBAMG and exposing their behavior as the problem size grows, the numerical experiments concentrate on Poisson-like scalar problems. Copyright

Adaptive Smoothed Aggregation in Lattice QCD

1 Department of Applied Mathematics, University of Colorado at Boulder, Email: brannick@newton.colorado.edu, mbrezina@math.cudenver.edu, maclachl@colorado.edu, tmanteuf@colorado.edu, stevem@colorado.edu, jruge@colorado.edu 2 Department of Mathematical Sciences, Ball State University, Email: ilivshits@bsu.edu 3 Department of Applied Physics and Applied Mathematics, Columbia University, Email: david.keyes@columbia.edu 4 Department of Mathematics, The Pennsylvania State University, Email: ludmil@psu.edu 5 SCI Institute, University of Utah, Email: livne@sci.utah.edu

A parallel version of a multigrid algorithm for isotropic transport equations

The focus of this paper is on a parallel algorithm for solving the transport equations in a slab geometry using multigrid. The spatial discretization scheme used is a finite element method called the modified linear discontinuous (MLD) scheme. The MLD scheme represents a lumped version of the standard linear discontinuous (LD) scheme. The parallel algorithm was implemented on the Connection Machine 2 (CM2). Convergence rates and timings for this algorithm on the CM2 and Cray-YMP are shown.

Extending the applicability of multigrid methods

Multigrid methods are ideal for solving the increasingly large-scale problems that arise in numerical simulations of physical phenomena because of their potential for computational costs and memory requirements that scale linearly with the degrees of freedom. Unfortunately, they have been historically limited by their applicability to elliptic-type problems and the need for special handling in their implementation. In this paper, we present an overview of several recent theoretical and algorithmic advances made by the TOPS multigrid partners and their collaborators in extending applicability of multigrid methods. Specific examples that are presented include quantum chromodynamics, radiation transport, and electromagnetics.