Modifying CLJP to select grid hierarchies with lower operator complexities and better performance

Alber, David M.

doi:10.1002/nla.473

Cited by 3 publications

(8 citation statements)

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“…In some cases, the operator complexity decreased by a small amount as the number of processors increased. Finally, as expected from previous observations [7,9,10], CLJP produces grid hierarchies with operator complexities so large the algorithm is not viable for this problem. CLJP produces unusually large operator complexities for problems on structured meshes, but this issue is not apparent with unstructured meshes.…”

Section: Tower Plotsmentioning

confidence: 56%

“…For some problems, such as problems on structured grids and problems in three dimensions, CLJP selects coarse-grid hierarchies with many more C-points than alternative algorithms, such as Falgout coarsening. Analysis of this behaviour is found in [10].…”

Section: End Formentioning

confidence: 99%

“…A modified form of CLJP, called CLJP in colour (CLJP-c), was first presented in [10]. CLJP-c modifies the initialization phase of CLJP to encourage the selection of a more structured coarse grid.…”

Section: Cljp In Colourmentioning

confidence: 99%

“…In [10], it was suggested that applying the graph colouring idea from CLJP-c to PMIS might yield positive results. Two new algorithms have come from this idea: PMIS-c1 and PMIS-c2.…”

Section: Pmis-c1 and Pmis-c2mentioning

confidence: 99%

“…The effectiveness of the AMG solve phase relies on the setup phase algorithms, which are responsible for selecting coarse grids and building grid transfer operators. A number of parallel coarse-grid selection algorithms have been developed [7][8][9][10]. The focus of this paper is on the efficiency and scalability of the setup phase algorithms and the quality of their results.…”

Section: Introductionmentioning

confidence: 99%

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Parallel coarse‐grid selection

Alber

Olson

2007

Numerical Linear Algebra App

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SUMMARYAlgebraic multigrid (AMG) is a powerful linear solver with attractive parallel properties. A parallel AMG method depends on efficient, parallel implementations of the coarse-grid selection algorithms and the restriction and prolongation operator construction algorithms. In the effort to effectively and quickly select the coarse grid, a number of parallel coarsening algorithms have been developed. This paper examines the behaviour of these algorithms in depth by studying the results of several numerical experiments. In addition, new parallel coarse-grid selection algorithms are introduced and tested.

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Section: Tower Plotsmentioning

confidence: 56%

Section: End Formentioning

confidence: 99%

Section: Cljp In Colourmentioning

confidence: 99%

“…In [10], it was suggested that applying the graph colouring idea from CLJP-c to PMIS might yield positive results. Two new algorithms have come from this idea: PMIS-c1 and PMIS-c2.…”

Section: Pmis-c1 and Pmis-c2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Parallel coarse‐grid selection

Alber

Olson

2007

Numerical Linear Algebra App

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Multigrid Methods

Mijalkovi��¹,

Mihajlovi��²

2008

Wiley Encyclopedia of Computer Science and Engineering

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Numerical modeling of important phenomena and processes in science and engineering has emerged in recent years as an alternative to the experimental approach. The ever‐increasing computer speed and storage capabilities enable modeling of large‐scale problems in continuum mechanics, electromagnetism, quantum physics, and other important areas. To perform the modeling task efficiently, sophisticated algorithms and numerical techniques for the solution of the underlying models are essential. Such models are expressed frequently as mathematical equations. To represent a continuous mathematical model on a computer, which operates on discrete binary‐coded data, a mathematical procedure referred to as the discretization needs to be deployed. A common result of the discretization procedure is a system of linear or non linear equations that needs to be solved. Numerical modeling of complex systems frequently requires the solution of large systems of equations (numbering several million). Such systems need to be solved repeatedly, as part of an iterative procedure, such as linearization, time‐stepping, optimization, or inverse problems. This reason is why the solution of linear systems is a crucial phase of the overall numerical modeling process. As a result, substantial research effort has been devoted to the development of effective numerical algorithms for the solution of linear systems. This article describes an important class of iterative algorithms for the solution of linear systems, referred to as multigrid (MG) methods. MG was developed originally in the context of the solution of discretized differential equations, with the aim to overcome the difficulties that standard iterative solvers encounter in this context. Over time, MG methods have evolved into an independent field of research, which has a major impact on many scientific and engineering disciplines. In this article, we present a brief review of MG and its applications to some realistic problems from numerical modeling practice.

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Algebraic interface‐based coarsening AMG preconditioner for multi‐scale sparse matrices with applications to radiation hydrodynamics computation

2016

Numerical Linear Algebra App

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Coarsening is a crucial component of algebraic multigrid (AMG) methods for iteratively solving sparse linear systems arising from scientific and engineering applications. Its application largely determines the complexity of the AMG iteration operator. Usually, high operator complexities lead to fast convergence of the AMG method; however, they require additional memory and as such do not scale as well in parallel computation. In contrast, although low operator complexities improve parallel scalability, they often lead to deterioration in convergence. This study introduces a new type of coarsening strategy called algebraic interface-based coarsening that yields a better balance between convergence and complexity for a class of multi-scale sparse matrices. Numerical results for various model-type problems and a radiation hydrodynamics practical application are provided to show the effectiveness of the proposed AMG solver. KEYWORDS algebraic multigrid (AMG), coarsening, preconditioning, parallel computation, multiscale sparse matrices, radiation hydrodynamics | INTRODUCTIONMany challenging computational problems in science and engineering need the solutions of large-scale sparse linear systems, that is,where A = (a ij ) n × n ∈ R n × n is a sparse matrix with entries a ij and u , f ∈ R n are the unknown and right-side vectors, respectively. The solver for these linear systems is a crucial component of simulation codes in diverse realistic applications and usually dominates the time consumption of simulations. Algebraic multigrid (AMG) 1-3 is one of the most efficient methods for iteratively solving linear systems arising from discretization of partial differential equations. In the past decades, challenges in real applications have sparked great interest in and further development of AMG methods. Many algorithm and theory variants have been developed to handle various problems (e.g., ). In particular, parallel algorithm designs and implementations of AMG have received considerable attention in the past 15 years as a means of scaling up to massively parallel machines, see, for example. [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47] Overall, AMG plays an increasingly important role in large-scale simulation for practical applications. 23,[48][49][50][51][52][53][54] The operator complexity (C op ) is a key performance measure for AMG methods. It is used as an important indicator of the computational cost of each iteration and storage requirements and is defined as follows:( 1:2) where A l denotes a matrix of level l and |A l | denotes the number of nonzero entries in A l . A 0 = A is the matrix of the finest * These authors contributed equally to this work.level (l = 0). The coarse-level matrix A l (l >0) usually comes from an application of the Galerkin approach (see Section 2). For large-scale computations on modern supercomputers, both low operator complexity and fast convergence are required. The classical AMG methods 3,32,39 are designed for fast convergence; however, they often...

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Modifying CLJP to select grid hierarchies with lower operator complexities and better performance

Cited by 3 publications

References 9 publications

Parallel coarse‐grid selection

Parallel coarse‐grid selection

Multigrid Methods

Algebraic interface‐based coarsening AMG preconditioner for multi‐scale sparse matrices with applications to radiation hydrodynamics computation

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