The development of an algebraic multigrid algorithm for symmetric positive definite linear systems

Vanek, P; Mandel, J; Brezina, M

Title: The development of an algebraic multigrid algorithm for symmetric positive definite linear systems

Full Record
Other Related Research

Abstract

An algebraic multigrid algorithm for symmetric, positive definite linear systems is developed based on the concept of prolongation by smoothed aggregation. Coarse levels are generated automatically. We present a set of requirements motivated heuristically by a convergence theory. The algorithm then attempts to satisfy the requirements. Input to the method are the coefficient matrix and zero energy modes, which are determined from nodal coordinates and knowledge of the differential equation. Efficiency of the resulting algorithm is demonstrated by computational results on real world problems from solid elasticity, plate blending, and shells.

Authors:

Vanek, P; Mandel, J; Brezina, M ^[1]

Univ. of Colorado, Denver, CO (United States)

Publication Date:: Tue Dec 31 00:00:00 EST 1996

Research Org.:: Front Range Scientific Computations, Inc., Lakewood, CO (United States)

OSTI Identifier:: 440740

Report Number(s):: CONF-9604167-Vol.2
ON: DE96015307; TRN: 97:000721-0062

Resource Type:: Conference

Resource Relation:: Conference: Copper Mountain conference on iterative methods, Copper Mountain, CO (United States), 9-13 Apr 1996; Other Information: PBD: [1996]; Related Information: Is Part Of Copper Mountain conference on iterative methods: Proceedings: Volume 2; PB: 242 p.

Country of Publication:: United States

Language:: English

Subject:: 99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; ALGORITHMS; DESIGN; CONVERGENCE; ITERATIVE METHODS; COORDINATES; NUMERICAL SOLUTION

Citation Formats


                    Vanek, P, Mandel, J, and Brezina, M. The development of an algebraic multigrid algorithm for symmetric positive definite linear systems.  United States: N. p., 1996. 
        Web.

Copy to clipboard


                    Vanek, P, Mandel, J, & Brezina, M. The development of an algebraic multigrid algorithm for symmetric positive definite linear systems.  United States.

Copy to clipboard


                    Vanek, P, Mandel, J, and Brezina, M. 1996.  
        "The development of an algebraic multigrid algorithm for symmetric positive definite linear systems".  United States.  https://www.osti.gov/servlets/purl/440740.

Copy to clipboard


                    
@article{osti_440740,

  title        = {The development of an algebraic multigrid algorithm for symmetric positive definite linear systems},

  author       = {Vanek, P and Mandel, J and Brezina, M},

  abstractNote = {An algebraic multigrid algorithm for symmetric, positive definite linear systems is developed based on the concept of prolongation by smoothed aggregation. Coarse levels are generated automatically. We present a set of requirements motivated heuristically by a convergence theory. The algorithm then attempts to satisfy the requirements. Input to the method are the coefficient matrix and zero energy modes, which are determined from nodal coordinates and knowledge of the differential equation. Efficiency of the resulting algorithm is demonstrated by computational results on real world problems from solid elasticity, plate blending, and shells.},

  doi          = {},

  url          = {https://www.osti.gov/biblio/440740},
  journal      = {},
number       = ,

  volume       = ,

  place        = {United States},

  year         = {Tue Dec 31 00:00:00 EST 1996},

  month        = {Tue Dec 31 00:00:00 EST 1996}

}

Copy to clipboard

Conference:

View Conference (14.94 MB)

Other availability

Please see Document Availability for additional information on obtaining the full-text document. Library patrons may search WorldCat to identify libraries that hold this conference proceeding.

Save / Share:

Export Metadata

Save to My Library

Similar records in OSTI.GOV collections:

Adaptive Algebraic Multigrid for Finite Element Elliptic Equations with Random Coefficients

Technical Report Kalchev, D

This thesis presents a two-grid algorithm based on Smoothed Aggregation Spectral Element Agglomeration Algebraic Multigrid (SA-{rho}AMGe) combined with adaptation. The aim is to build an efficient solver for the linear systems arising from discretization of second-order elliptic partial differential equations (PDEs) with stochastic coefficients. Examples include PDEs that model subsurface flow with random permeability field. During a Markov Chain Monte Carlo (MCMC) simulation process, that draws PDE coefficient samples from a certain distribution, the PDE coefficients change, hence the resulting linear systems to be solved change. At every such step the system (discretized PDE) needs to be solved and themore »« less
https://doi.org/10.2172/1047794

Full Text Available
Scalability and performance of data-parallel pressure-based multigrid methods for viscous flows

Journal Article Blosch, E; Shyy, W - Journal of Computational Physics

A full-approximation storage multigrid method for solving the steady-state 2-d incompressible Navier-Stokes equations on staggered grids has been implemented in Fortran on the CM-5, using the array aliasing feature in CM-Fortran to avoid declaring fine-grid-sized arrays on all levels while still allowing a variable number of grid levels. Thus, the storage cost scales with the number of unknowns, allowing us to consider significantly larger problems than would otherwise be possible. Timings over a range of problem sizes and numbers of processors, up to 4096 x 4096 on 512 nodes, show that the smoothing procedure, a pressure-correction technique, is scalable andmore »« less
https://doi.org/10.1006/jcph.1996.0098
Reducing Parallel Communication in Algebraic Multigrid through Sparsification

Journal Article Bienz, Amanda; Falgout, Robert; Gropp, William; ... - SIAM Journal on Scientific Computing

Algebraic multigrid (AMG) is an $$\mathcal{O}(n)$$ solution process for many large sparse linear systems. A hierarchy of progressively coarser grids which utilize complementary relaxation and interpolation operators is constructed. High-energy error is reduced by relaxation, while low-energy error is mapped to coarse-grid matrices and reduced there. However, large parallel communication costs often limit parallel scalability. Furthermore, as the multigrid hierarchy is formed, each coarse matrix is formed through a triple matrix product. The resulting coarse grids often have significantly more nonzeros per row than the original fine-grid operator, thereby generating high parallel communication costs associated with sparse matrix-vector multiplication (SpMV)more »« less
https://doi.org/10.1137/15m1026341

Full Text Available
Coarse-grid selection for parallel algebraic multigrid

Conference

The need to solve linear systems arising from problems posed on extremely large, unstructured grids has sparked great interest in parallelizing algebraic multigrid (AMG) To date, however, no parallel AMG algorithms exist We introduce a parallel algorithm for the selection of coarse-grid points, a crucial component of AMG, based on modifications of certain paallel independent set algorithms and the application of heuristics 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 effect on multigrid convergence, and AMG complexity
Full Text Available
A Generalized Eigensolver based on Smoothed Aggregation (GES-SA) for Initializing Smoothed Aggregation Multigrid (SA)

Journal Article Brezina, M; Manteuffel, T; McCormick, S; ... - Numerical Linear Algebra with Applications, vol. 15, no. 2-3, January 15, 2008, pp. 249-269

Consider the linear system Ax = b, where A is a large, sparse, real, symmetric, and positive definite matrix and b is a known vector. Solving this system for unknown vector x using a smoothed aggregation multigrid (SA) algorithm requires a characterization of the algebraically smooth error, meaning error that is poorly attenuated by the algorithm's relaxation process. For relaxation processes that are typically used in practice, algebraically smooth error corresponds to the near-nullspace of A. Therefore, having a good approximation to a minimal eigenvector is useful to characterize the algebraically smooth error when forming a linear SA solver. Thismore »« less
Full Text Available

Similar Records