Energy Optimization of Algebraic Multigrid Bases

Brezina, Marian; Mandel, Jan; Vaněk, Petr

doi:10.1007/s006070050022

Cited by 126 publications

(135 citation statements)

References 0 publications

Supporting

Mentioning

131

Contrasting

Unclassified

Order By: Relevance

“…The proposed prolongation smoothing algorithm is an extension of [2] to the indefinite, non-Hermitian case through the use of the A * A norm. Moreover, a constrained Krylov method is used as the minimization algorithm.…”

Section: Discussionmentioning

confidence: 99%

“…SA [1][2][3] style AMG [4,5] is a popular and effective iterative solver and preconditioner, yet there are a number of robustness issues associated with this algorithm. For wave-like problems, e.g.…”

Section: Smoothed Aggregationmentioning

confidence: 99%

“…In this section, we present an overview of the SA setup algorithm [1][2][3]. The function of the SA setup phase is to construct coarse operators A k , with A 0 = A being the index associated with the finest level, through the construction of interpolation operators P k : R n k+1 → R n k , where n k and n k+1 are sizes of two successively coarser grids.…”

Section: Sa Algorithmmentioning

confidence: 99%

“…A discussion of coarsening is omitted since aggregation generally results in groupings of three fine degrees-of-freedom per aggregate as expected in 1D. We first propose an effective prolongation smoother for non-Hermitian and indefinite problems based on a variant of the energy-minimizing prolongation smoother [2], but posed in the A * A norm. Specifically, we minimize the energy for each of the n columns of P with…”

Section: Solver Componentsmentioning

confidence: 99%

See 3 more Smart Citations

Smoothed aggregation for Helmholtz problems

Olson

Schroder

2010

Numerical Linear Algebra App

View full text Add to dashboard Cite

SUMMARYWe outline a smoothed aggregation algebraic multigrid method for 1D and 2D scalar Helmholtz problems with exterior radiation boundary conditions. We consider standard 1D finite difference discretizations and 2D discontinuous Galerkin discretizations. The scalar Helmholtz problem is particularly difficult for algebraic multigrid solvers. Not only can the discrete operator be complex-valued, indefinite, and non-self-adjoint, but it also allows for oscillatory error components that yield relatively small residuals. These oscillatory error components are not effectively handled by either standard relaxation or standard coarsening procedures. We address these difficulties through modifications of SA and by providing the SA setup phase with appropriate wave-like near null-space candidates. Much is known a priori about the character of the near null-space, and our method uses this knowledge in an adaptive fashion to find appropriate candidate vectors. Our results for GMRES preconditioned with the proposed SA method exhibit consistent performance for fixed points-per-wavelength and decreasing mesh size.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Smoothed Aggregationmentioning

confidence: 99%

Section: Sa Algorithmmentioning

confidence: 99%

Section: Solver Componentsmentioning

confidence: 99%

See 2 more Smart Citations

Smoothed aggregation for Helmholtz problems

Olson

Schroder

2010

Numerical Linear Algebra App

View full text Add to dashboard Cite

show abstract

“…However, they are built on several heuristics and so a rigorous analysis of their coefficient-robustness is difficult (see [22] for a review of existing theoretical results). Nevertheless, the key principle of these algebraic coarse spaces, namely energy minimisation [11], also underlies many other coarse spaces. To obtain rigorous coefficient-independent convergence results we will need to work in the following energy and weighted…”

Section: Introductionmentioning

confidence: 99%

Robust Coarsening in Multiscale PDEs

Scheichl

2013

Lecture Notes in Computational Science and Engineering

View full text Add to dashboard Cite

Multigrid Methods forFEMandBEMApplications

Hackbusch

2004

Encyclopedia of Computational Mechanics

View full text Add to dashboard Cite

After discretisation of the partial differential equations from mechanics one usually obtains large systems of (non)linear equations. Their efficient solution requires the use of fast iterative methods. Multi-grid iterations are able to solve linear and nonlinear systems with a rather fast rate of convergence, provided the problem is of elliptic type. The contribution describes the basic construction of multigrid methods, their ingredients, related methods, and their application to various problem classes. Although most of the applications concern FEM discretisations of partial differential equations, there are also applications to integral equations as they occur in boundary element methods (BEM).

show abstract

Energy Optimization of Algebraic Multigrid Bases

Cited by 126 publications

References 0 publications

Smoothed aggregation for Helmholtz problems

Smoothed aggregation for Helmholtz problems

Robust Coarsening in Multiscale PDEs

Multigrid Methods forFEMandBEMApplications

Contact Info

Product

Resources

About