Black box multigrid

Dendy, J.E.

doi:10.1016/0021-9991(82)90057-2

Cited by 310 publications

(231 citation statements)

References 7 publications

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“…The solution to the problem with large jumps in coefficients, first discussed in [2] and further developed in [8,9], is to allow the coefficients of the interpolation operator, Z, to depend on the coefficients of the matrix, A. Such an operator-induced interpolation is better able to reflect the slow-to-converge errors of relaxation as these errors are, themselves, dependent on the variation in A.…”

Section: The Black Box Multigrid Methodsmentioning

confidence: 99%

“…The black box multigrid technique, first introduced in [2], uses geometrically structured coarse grids in combination with an interpolation operator designed to account for the effects of jumps in the diffusion coefficients to achieve fast multigrid convergence in many situations [4,8,10]. Algebraic multigrid, or AMG, is also known to be effective for elliptic problems with jumps in their coefficients [33,36], achieving this efficiency by tailoring both the coarse-grid structure and interpolation operator to account for the jumps in the coefficients.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Fast and robust solvers for pressure-correction in bubbly flow problems

MacLachlan

Tang

Vuik

2008

Journal of Computational Physics

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Section: The Black Box Multigrid Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fast and robust solvers for pressure-correction in bubbly flow problems

MacLachlan

Tang

Vuik

2008

Journal of Computational Physics

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“…For some comparative experiments, see [18,19]. Some easily accessible publications in which Gslerkin coarse grid approximation is used are [l, 5,6,10,11,16,17]. For a justification of the appellation "Galerkin co~ree grid approximation", and for a few remarks on efficient programming of r A"'"pk, see [18] and [19] respectively• 6.…”

Section: S Coarse Grid Approximationmentioning

confidence: 99%

“…Details are given of a FORTRAN code called MGDlV; see also [18,19]. MGDlV is an example of what has been called "black box" multigrid method in [5]; there another example of such a method is presented, which is particularly suited for problems with discontinuous coefficients. A similar method has been developed in [9,10,11].…”

Section: Introductionmentioning

confidence: 99%

Multigrid methods: development of fast solvers

Hemker¹,

Kettler

Wesseling

et al. 1983

Applied Mathematics and Computation

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Numerical and programming aspects are discussed of multigrid algorithms for the solution of discretized linear elliptic equations. The aim is to obtain software that is perceived and can be used just like any standard subroutine for solving systems of linear equations. The user has to specify only the matrix and the right-hand-side, and remains unaware of the underlying multigrid method. We find that a large class of equations can be solved efficiently in this way. The equation may be non-self-adjoint, and its coefficients are arbitrary. Special attention is given to the treatment of the convection-diffusion equation at high P~clet number. Details are given of an available portable FORTRAN code, which vectorizes satisfactorily on vector machines. CP time statistics are given for a CYBER-205.

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“…Unfortunately, bilinear interpolation causes smoothing of coarse solutions across object boundaries (represented by small diffusion constants) in an image, causing a poor convergence rate. In contrast, algebraic multigrid [21,22] uses the diffusion constants to generate problem-specific interpolation operators and coarsened matrices. Unfortunately, the coarsened matrices produced via algebraic multigrid are not guaranteed to represent a banded (lattice) sparsity structure.…”

Section: Introductionmentioning

confidence: 99%

A Lattice-Preserving Multigrid Method for Solving the Inhomogeneous Poisson Equations Used in Image Analysis

Grady

2008

Lecture Notes in Computer Science

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Abstract. The inhomogeneous Poisson (Laplace) equation with internal Dirichlet boundary conditions has recently appeared in several applications ranging from image segmentation [1-3] to image colorization [4], digital photo matting [5,6] and image filtering [7,8]. In addition, the problem we address may also be considered as the generalized eigenvector problem associated with Normalized Cuts [9], the linearized anisotropic diffusion problem [10,11,8] solved with a backward Euler method, visual surface reconstruction with discontinuities [12,13] or optical flow [14]. Although these approaches have demonstrated quality results, the computational burden of finding a solution requires an efficient solver. Design of an efficient multigrid solver is difficult for these problems due to unpredictable inhomogeneity in the equation coefficients and internal Dirichlet boundary conditions with unpredictable location and value. Previous approaches to multigrid solvers have typically employed either a data-driven operator (with fast convergence) or the maintenance of a lattice structure at coarse levels (with low memory overhead). In addition to memory efficiency, a lattice structure at coarse levels is also essential to taking advantage of the power of a GPU implementation [15,16,5,3]. In this work, we present a multigrid method that maintains the low memory overhead (and GPU suitability) associated with a regular lattice while benefiting from the fast convergence of a data-driven coarse operator.

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Black box multigrid

Cited by 310 publications

References 7 publications

Fast and robust solvers for pressure-correction in bubbly flow problems

Fast and robust solvers for pressure-correction in bubbly flow problems

Multigrid methods: development of fast solvers

A Lattice-Preserving Multigrid Method for Solving the Inhomogeneous Poisson Equations Used in Image Analysis

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