A. Brandt, S. McCormick, and J. Ruge. Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations. Technical report, Institute for Computational Studies, Fort Collins, CO, POB 1852, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and give theoretical justification for their use as tools for measuring the quality of coarse grids. Key words. algebraic multigrid, compatible relaxation 1. Introduction. The algebraic multigrid (AMG) method was originally developed to solve general matrix equations using multigrid principles [7, 3, 8, 19, 4, 20]. The fact that it used only information in the underlying matrix made it attractive as a potential black box solver, a notion that has since been all but abandoned. Instead, a wide variety of AMG algorithms have been developed that target di#erent problem classes and have di#erent robustness and ....

A. Brandt, S. F. McCormick, and J. W. Ruge, Algebraic multigrid (AMG) for automatic multigrid solutions with application to geodetic computations. Report, Inst. for Computational Studies, Fort Collins, Colo., October 1982.

....was performed under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National Laboratory under contract No. W 7405 Eng 48. Preprint submitted to Elsevier Preprint 5 December 2000 1 Introduction Algebraic multigrid (AMG) was introduced in the 1980 s [3,1,2,4,18,14,16,15] and immediately attracted the attention of scientists needing to solve large problems posed on unstructured grids. In the past several years there has been a major surge of interest in AMG. Much of the current research focuses either on improving the standard AMG algorithm [9,8] or on dramatic ....

.... to points in C i that are connected to e k (hence the C1 requirement) This yields the formula for the interpolation weights, w i;j = 1 a i;j X k2D w i a i;k 0 B B B a i;j X k2D s i a i;k a k;j X m2C i a k;m 1 C C C A : 7) AMG theory was developed originally [3,15] under the assumption that the operator matrix A is an M matrix, and that the non zero o diagonal entries are all of the opposite sign as the diagonal entries (which all have the same sign) While this assumption is necessary for convergence proof, AMG works 14 quite well on matrices that do not ....

A. Brandt, S. F. McCormick, and J. W. Ruge, Algebraic multigrid (AMG) for automatic multigrid solutions with application to geodetic computations. Report, Inst. for Computational Studies, Fort Collins, Colo., October 1982.

....designed to insure the quality of the coarse grids. A prototype serial version of the algorithm is implemented, and tests are conducted to determine its e ect on multigrid convergence, and AMG complexity. 1 Introduction Since the introduction of algebraic multigrid (AMG) in the 1980 s [4, 2, 3, 5, 19, 16, 18, 17] the method has attracted the attention of scientists needing to solve large problems posed on unstructured grids. Recently, there has been a major surge of interest in the eld, due in large part to the need to solve increasingly larger systems, with hundreds of millions or billions of unknowns. ....

A. Brandt, S. F. McCormick, and J. W. Ruge, Algebraic multigrid (AMG) for automatic multigrid solutions with application to geodetic computations. Report, Inst. for Computational Studies, Fort Collins, Colo., October 1982.

....methods. Key words. algebraic multigrid, incomplete LU factorization, multigraph methods. AMS subject classi cations. 65M55, 65N55 1. Introduction. In this work, we develop a multilevel multigraph algorithm. Algebraic multigrid methods are currently a topic of intense research interest [17, 18, 20, 46, 12, 48, 38, 11, 44, 3, 4, 1, 2, 5, 16, 7, 29, 28, 27, 42, 41, 21]. An excellent recent survey is given in Wagner [49] In many real world calculations, direct methods are still widely used [6] The robustness of direct elimination methods and their simplicity of use often outweigh the apparent bene ts of fast iterative solvers. Our goal here is to try to ....

....For matrices arising from discretizations of partial di erential equations, often the sparsity of the matrix A is related in some way to the underlying grid, and the problem of coarsening the graph 10 of the matrix A can be formulated in terms of coarsening the grid. Some examples are given in [14, 13, 17, 18, 46, 12, 49]. In this case, one has the geometry of the grid to serve as an aid in developing and analyzing the coarsening procedure. There are also more general graph coarsening algorithms [32, 33, 19] often used to partition problems for parallel computation. Here our coarsening scheme is based upon ....

A. Brandt, S. McCormick, and J. Ruge, Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations, tech. rep., Institute for Computational Studies, Colorado State University, Fort Collins CO, 1982.

....algorithm will eventually provide reasonably good rates of convergence for many classes of problems, while requiring only minimal input. Algebraic approaches to multilevel methods have enjoyed a long history, beginning with the algebraic multigrid (AMG) methods of Brandt, McCormick, and Ruge [13, 14] Ruge and Stuben [26] and the black box multigrid method of Dendy [15] More recent work can be found in [1, 3, 4, 12, 20, 19, 17] as well as many contributions in [2] Our work grew out of the grid coarsening schemes developed in [10, 11] and the corresponding hierarchical basis iterations, ....

A. Brandt, S. McCormick, and J. Ruge, Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations, tech. report, Institute for Computational Studies, Colorado State University, Fort Collins CO, 1982.

.... approaches by de Zeeuw [115] and Reusken [85, 86] Other related approaches include frequency decomposition by Hackbusch [60] and filtering decomposition by Wittum [108, 109] The purely algebraic methods, on the other end of the spectrum, were first proposed by Brandt, McCormick and Ruge [25], and then popularized by Ruge and Stuben [88] These is a recent resurgence of interest in AMG and other multigrid algorithms with focuses on parallel implementation and memory hierarchy aspects [26, 37, 38, 44, 45, 70, 84, 97] An introduction to AMG is recently given by Stuben [94] See also ....

....of the PDE coefficients. For example, multigrid converges slowly when the coefficients exhibit anisotropy [59] large jumps in discontinuity [1, 19, 39, 40] or large oscillations [48, 73, 95] Special techniques such as line Gauss Seidel [19] semi coarsening [41, 42, 90] algebraic multigrid [14, 25, 85, 88, 93], frequency decomposition [43, 60, 95] and homogenization [48, 73] are used to handle some of these cases. In the next sections, we survey the state of the art of each individual multigrid components and discuss how they bring insight into the design of robust multigrid methods. 2. ....

[Article contains additional citation context not shown here]

A. Brandt, S. McCormick, and J. W. Ruge. Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations. Technical report, Inst. for Computational Studies, Fort Collins, CO, 1982.

No context found.

A. Brandt, S. McCormick, and J. Ruge. Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations. Technical report, Institute for Computational Studies, Fort Collins, CO, POB 1852, 1982.

No context found.

A. Brandt, S. McCormick, J. Rudge, Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations, Institute for Computational Studies, POB 1852, Fort Collins, Colorado, 1982.

....stopping criterion http: www.llnl.gov CASC and tolerance 10 9 . All computations were performed on a SGI ORIGIN 2000 computer with 8 MIPS R10000 processors running at 250MHz with 4MB L2 cache and total main memory of 4GB. The algebraic multigrid (AMG) was first introduced in the 1980 s [1, 2, 9]. The main advantage of the AMG algorithm when compared to standard multigrid is that it does not require access to any additional information (such as hierarchy of grids, coarse grid and interpolation matrices) the algorithm constructs them itself. Thus, it is applicable to linear systems ....

A. Brandt, S. F. McCormick, and J. W. Ruge, Algebraic multigrid (AMG) for automatic multigrid solutions with application to geodetic computations, Report, Inst. for Computational Studies, Fort Collins, Colo., October 1982.

....by abstracting, in an algebraic sense, the properties that make geometric multigrid methods e#ective. Ideally, this results in a method that is automatic and robust. The classical formulation of AMG grew from the e#orts of Brandt, McCormick, Ruge, and Stuben in the 1980 s. For details, see [2, 1, 17]. Interest in AMG methods is growing rapidly, both in academia and in industry, because these methods have great potential for solving the large scale problems common to many modern applications. AMG has proved to be e#ective on a large class of problems (see, e.g. 8, 17] especially scalar ....

A. Brandt, S. McCormick, and J. Ruge, Algebraic multigrid (AMG) for automatic multigrid solutions with application to geodetic computations, tech. report, Inst. for Computational Studies, Fort Collins, Colorado, October 1982.

....c were used, and the exact values of the other unknowns and the right hand sides were defined consistently. The minimization problems were discretized using bilinear finite elements on a uniform square mesh, with h ranging from 1 4 to 1 128 . For simplicity, we used Algebraic Multigrid (AMG [5]) to solve the discrete systems, with separate coarse grids and intergrid transfer operators determined for each unknown. This is not a block relaxation scheme because, while coarsening is set up within each level, all variables are coupled in the coarse grid correction process. We started with ....

A. Brandt, S. McCormick, and J. Ruge, Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations, report, Inst. Comp. Studies, Colo. State Univ., l982.

....unstructured grids. Such large grid simulations require the e#cient union of massively parallel computing with scalable numerical algorithms such as multigrid (see e.g. 2] An especially e#ective method for many of the problems that arise in these applications is algebraic multigrid (AMG [6, 4, 5, 7, 21, 18, 20, 19]) AMG is a method for solving matrix equations that is based on multigrid concepts, but constructs the coarsening process in an algebraic way that requires no explicit knowledge of the geometry. It examines the matrix entries to determine a sequence of smaller matrix problems that serve as ....

....case, the interpolation matrix, P , must be defined so that algebraically smooth error is e#ectively eliminated in step (2.2b) and the coarse grid equations, which involve P T AP , are amenable to solution. 2.1. AMG. To define the multigrid components in AMG, we use the following heuristic (cf. [6, 4, 19]) based on special properties of M matrices: H2: Smooth error varies slowest in the direction of strong dependence. Here, we say that unknown i strongly depends on unknown j if a i,j # # max k #=i a i,k , for some fixed # # (0, 1) 2.3) Thus, strong dependence is characterized ....

A. Brandt, S. F. McCormick, and J. W. Ruge, Algebraic multigrid (AMG) for automatic multigrid solutions with application to geodetic computations. Report, Inst. for Computational Studies, Fort Collins, Colo., October 1982.

....convergence rate of AMG is comparable with geometric multigrid methods although it can not be proved in general. Applications of AMG methods in various practical (engineering) areas are given in [12, 13, 18, 19, 20] The first serious approach to AMG was made 1982 by Brandt, McCormick and Ruge in [5] and an improved version of it can be found in [6] This method is mainly concerned with SPD matrices K h , which are additionally M matrices. In this approach the smoother is usually fixed (e.g. Gauss Seidel point relaxation) and the prolongation operator is constructed such that the error which ....

A. Brandt, S. McCormick, and J. W. Ruge, Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations, Report, Inst. Comp. Studies Colorado State Univ., 1982.

....unstructured grids. Such large grid simulations require the efficient union of massively parallel computing with scalable numerical algorithms such as multigrid (see e.g. 2] An especially effective method for many of the problems that arise in these applications is algebraic multigrid (AMG [6, 4, 5, 7, 21, 18, 20, 19]) AMG is a method for solving matrix equations that is based on multigrid concepts, but constructs the coarsening process in an algebraic way that requires no explicit knowledge of the geometry. It examines the matrix entries to determine a sequence of smaller matrix problems that serve as ....

....the interpolation matrix, P , must be defined so that algebraically smooth error is effectively eliminated in step (2.2b) and the coarse grid equations, which involve P T AP , are amenable to solution. 2.1. AMG. To define the multigrid components in AMG, we use the following heuristic (cf. [6, 4, 19]) based on special properties of M matrices: H2: Smooth error varies slowest in the direction of strong dependence. Here, we say that unknown i strongly depends on unknown j if Gammaa i;j max k 6=i f Gammaa i;k g; for some fixed 2 (0; 1) 2.3) Thus, strong dependence is characterized ....

A. Brandt, S. F. McCormick, and J. W. Ruge, Algebraic multigrid (AMG) for automatic multigrid solutions with application to geodetic computations. Report, Inst. for Computational Studies, Fort Collins, Colo., October 1982.

....in which standard AMG does not work well, and indicate the current directions taken by AMG researchers to alleviate these diculties. Key words. algebraic multigrid, interpolation, unstructured meshes, scalability 1. Introduction. Algebraic multigrid (AMG) was rst introduced in the early 1980 s [11, 8, 10, 12], and immediately attracted substantial interest [32, 28, 30, 29] Research continued at a modest pace into the late 1980 s and early 1990 s [18, 14, 21, 25, 20, 26, 22] Recently, however, there has been a major resurgence of interest in the eld, for classical AMG as de ned in [29] as well as ....

A. Brandt, S. F. McCormick, and J. W. Ruge, Algebraic multigrid (AMG) for automatic multigrid solutions with application to geodetic computations. Report, Inst. for Computational Studies, Fort Collins, Colo., October 1982.

....As a result, codes are being developed to solve complex multi physics problems on highly resolved, unstructured grids. Such large grid simulations often require massively parallel computing as well as scalable numerical algorithms such as multigrid (see e.g. 1] Algebraic Multigrid (AMG) [5, 3, 4, 6, 19, 16, 18, 17] is a method for solving matrix equations that is based on multigrid concepts. It examines the matrix entries to determine a sequence of smaller matrix problems that serve as coarse level equations. AMG also determines associated interlevel transfer operators (restriction and prolongation) then ....

A. Brandt, S. F. McCormick, and J. W. Ruge, Algebraic multigrid (AMG) for automatic multigrid solutions with application to geodetic computations. Report, Inst. for Computational Studies, Fort Collins, Colo., October 1982.

.... Omega Gamma we refer to [11, 3, 1, 16, 20] An algebraic multigrid approach for the solution of (1) requires in addition to the available components of the geometric multigrid also a proper coarsening strategy. In spite of the fact that the FE matrix K e h is SPD, the classical approaches of [5, 6, 7, 8, 15, 18, 19, 22] and variants of it fail for the problem at hand. All these methods are designed for SPD problems which either stems from an FE discretization for H 1 elliptic problems or needs beside the SPD property special characteristics of the system matrix (e.g. M matrix property) A first AMG approach ....

A. Brandt, S. McCormick, and J. W. Ruge, Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations, Report, Inst. Comp. Studies Colorado State Univ., 1982.

....As a result, codes are being developed to solve complex multi physics problems on highly resolved, unstructured grids. Such large grid simulations often require massively parallel computing as well as scalable numerical algorithms such as multigrid (see e.g. 1] Algebraic Multigrid (AMG) [5, 3, 4, 6, 16, 13, 15, 14] is well suited for solving unstructured grid problems, and works remarkably well over a wide variety of applications (see, e.g. 7] However, for some problems, AMG is not e ective without certain problem speci c modi cations or careful parameter tuning. There are still other problems (e.g. ....

A. Brandt, S. F. McCormick, and J. W. Ruge, Algebraic multigrid (AMG) for automatic multigrid solutions with application to geodetic computations. Report, Inst. for Computational Studies, Fort Collins, Colo., October 1982.

....energy minimization problem is somewhat similar to the minimization problem encountered in structural mechanics, for which very efficient multigrid solvers have been developed. Of these, the closest to the ones needed in molecular mechanics are the algebraic multigrid (AMG) solvers [12, 13.1] [36], 38] 14] 73] which do not assume that the problem arises from PDE or that the unknowns are really placed on a grid. The methods we have developed for molecular energy minimization follow the general AMG outline: coarser levels are constructed each by taking a suitable subset of the ....

....approximately the same electric potential under whatever input currents applied to the network. The first possible role for a multiscale approach here is in terms of a fast solver for such networks. Since the network is highly disordered, algebraic multigrid (AMG) solvers best fit the task (see [36], 38] 14] 73] As pointed out by Sorin Solomon (in a private letter to us) there is in fact no need to solve the electric network problem for any particular input currents: The coarse level nodes defined by the AMG coarsening process can directly be identified with the desired picture ....

A. BRANDT, S. MCCORMICK AND J. RUGE, Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations, Institute for Computational Studies, POB 1852, Fort Collins, Colorado, 1982.

....Dirac equations. AMS subject classifications. 35A40, 65F10, 65K10, 65M55, 65N22, 65N55, 65Y05, 76M20. 1. Introduction. Algebraic multigrid (AMG) algorithms are solvers of linear systems of equations which are based on multigrid principles but do not explicitly use the geometry of grids; see [20], 8] 32] 36] The emphasis in AMG is on automatic procedures for coarsening the set of equations, relying exclusively on its algebraic relations. AMG is widely employed for solving discretized partial differential equations (PDEs) on unstructured grids, or even on structured grids when the ....

....on its algebraic relations. AMG is widely employed for solving discretized partial differential equations (PDEs) on unstructured grids, or even on structured grids when the coarse grid can no longer be structured, or when the PDE has highly disordered coefficients. AMG can also be used (as in [20]) for certain discrete systems not arising from differential equations. The scope of AMG solvers has been rather limited, though. Its coarsening procedures have been inadequate for general non scalar, or high order, or non elliptic and anisotropic PDE systems, and also for non variational ....

[Article contains additional citation context not shown here]

A. Brandt, S. McCormick and J. Ruge, Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations, Institute for Computational Studies, POB 1852, Fort Collins, Colorado, 1982.

....the molecular energy minimization problem is somewhat similar to the minimization problem encountered in structural mechanics, for which very efficient multigrid solvers have been developed. Of those, the closest to the ones needed in molecular mechanics are the algebraic multigrid (AMG) solvers [A1] [A4] which do not assume that the problem arises from a continuum structure 5 or that the unknowns are really placed on a grid. Some of the basic ideas of the AMG solvers, as well as the experience obtained with them, will be important in our present development. 3.1 Main difficulty and ....

A. Brandt, S. McCormick and J. Ruge, Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations, Institute for Computational Studies, POB 1852, Fort Collins, Colorado, 1982.

....by recombining iterants. An application to the highly disordered Dirac equations is briefly reviewed. 1. Introduction. Algebraic multigrid (AMG) algorithms are solvers of linear systems of equations which are based on multigrid principles but do not explicitly use the geometry of grids; see [20], 8] 32] 36] The emphasis in AMG is on automatic procedures for coarsening the set of equations, relying exclusively on its algebraic relations. AMG is widely employed for solving discretized partial differential equations (PDEs) on unstructured grids, or even on structured grids when the ....

....on its algebraic relations. AMG is widely employed for solving discretized partial differential equations (PDEs) on unstructured grids, or even on structured grids when the coarse grid can no longer be structured, or when the PDE has highly disordered coefficients. AMG can also be used (as in [20]) for certain discrete systems not arising from differential equations. The scope of AMG solvers has been rather limited, though. Its coarsening procedures have been inadequate for general non scalar, or high order, or non elliptic and anisotropic PDE systems, and also for non variational ....

[Article contains additional citation context not shown here]

A. Brandt, S. McCormick and J. Ruge, Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations, Institute for Computational Studies, POB 1852, Fort Collins, Colorado, 1982.

....highly accurate, as needed for other applications (e.g. once for all derivation of macroscopic equations) 1. Introduction. Algebraic multigrid (AMG) algorithms are solvers of linear systems of equations which are based on multigrid principles but do not explicitly use the geometry of grids; see [15], 5] 23] 26] The emphasis in AMG is on automatic procedures for coarsening the set of equations, relying exclusively on its algebraic relations. AMG is widely employed for solving discretized partial differential equations (PDEs) on unstructured grids, or even on structured grids when the ....

....on its algebraic relations. AMG is widely employed for solving discretized partial differential equations (PDEs) on unstructured grids, or even on structured grids when the coarse grid can no longer be structured, or when the PDE has highly disordered coefficients. AMG can also be used (as in [15]) for certain discrete systems not arising from differential equations. The scope of AMG solvers has been rather limited, though. Its coarsening procedures have been inadequate for general non scalar, or high order, or non elliptic and anisotropic PDE systems, and also for non variational ....

[Article contains additional citation context not shown here]

A. Brandt, S. McCormick and J. Ruge, Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations, Institute for Computational Studies, POB 1852, Fort Collins, Colorado, 1982.

No context found.

A. Brandt, S. McCormick, and J. W. Ruge. Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations. Technical report, Inst. for Computational Studies, Fort Collins, CO, 1982.

No context found.

A. Brandt, S. McCormick, and J. Ruge. Algebraic multigrid (AMG) for automatic multigrid solutions with applications to geodetic computations. Technical report, Institute for Computational Studies, Fort Collins, Colorado, 1982.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC