A New Petrov–Galerkin Smoothed Aggregation Preconditioner for Nonsymmetric Linear Systems

Sala, Marzio; Tuminaro, Raymond S.

doi:10.1137/060659545

Cited by 68 publications

(103 citation statements)

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“…This enables approximate solvers (e.g., AMG [35,39,42]) to be used in conjunction with BDD methods. Although this approach relaxes the arithmetic/memory demands of sparse direct solvers (particularly in 3D), it would result in a new source of difficulties (e.g., robustness/complexity trade-off evaluation, parameter tuning, load unbalancing issues, poor flop rates) that deserve further research.…”

Section: 4mentioning

confidence: 99%

Implementation and Scalability Analysis of Balancing Domain Decomposition Methods

Badia

Martı́n

Principe

2013

Arch Computat Methods Eng

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Abstract. In this paper we present a detailed description of a high-performance distributedmemory implementation of balancing domain decomposition preconditioning techniques. This coverage provides a pool of implementation hints and considerations that can be very useful for scientists that are willing to tackle large-scale distributed-memory machines using these methods. On the other hand, the paper includes a comprehensive performance and scalability study of the resulting codes when they are applied for the solution of the Poisson problem on a large-scale multicore-based distributed-memory machine with up to 4096 cores. Well-known theoretical results guarantee the optimality (algorithmic scalability) of these preconditioning techniques for weak scaling scenarios, as they are able to keep the condition number of the preconditioned operator bounded by a constant with fixed load per core and increasing number of cores. The experimental study presented in the paper complements this mathematical analysis and answers how far can these methods go in the number of cores and the scale of the problem to still be within reasonable ranges of efficiency on current distributed-memory machines. Besides, for those scenarios where poor scalability is expected, the study precisely identifies, quantifies and justifies which are the main sources of inefficiency.Key words. Domain decomposition, parallelization, scalability, coarse-grid correction, balancing domain decomposition, BNN, BDDC AMS subject classifications. 65N55, 65F08, 65N30, 65Y05, 65Y201. Introduction. Scientific phenomena governed by partial differential equations (PDEs) can range from solid mechanics to fluid mechanics and electrodynamics, including any of the possible couplings. The solution of these equations can be approximated with the aid of computers by a discretization (and possibly linearization) and the subsequent numerical solution of the resulting sparse set of linear equations. This work is concerned with the fast solution of the Poisson problem discretized by the finite element (FE) method. Although the Poisson problem is the simplest model problem for, e.g., fluid flow simulation, it is still very useful as a building block for the "physics-based" preconditioning of very complex scientific applications governed by coupled systems of PDEs [1].The ever increasing demand of reality in the simulation of the complex scientific and engineering three-dimensional (3D) problems faced nowadays ends up with the solution of very large and sparse linear systems with several hundreds and even thousands of millions of equations/unknowns. The solution of these systems in a moderate time requires the vast amount of computational resources provided by current multicore-based distributed-memory machines. It is therefore essential to design parallel algorithms able to take profit of their underlying architecture.

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Section: 4mentioning

confidence: 99%

Implementation and Scalability Analysis of Balancing Domain Decomposition Methods

Badia

Martı́n

Principe

2013

Arch Computat Methods Eng

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“…Petrov-Galerkin coarsening [8][9][10], i.e. R =P * , is desirable for AMG when applied to non-Hermitian matrices.…”

Section: E N T a T I V E P R O L O N G A T O R 36mentioning

confidence: 99%

Smoothed aggregation for Helmholtz problems

Olson

Schroder

2010

Numerical Linear Algebra App

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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.

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“…In [12] aggregation-based AMG methods are investigated for nonsymmetric fluid dynamics problems, where the importance of the choice of transfer operators is emphasized. Multigrid methods can typically not be applied to the complete linear system, though, making the use of segregated preconditioners necessary.…”

Section: ) Solve Phasementioning

confidence: 99%

Design of a Parallel Hybrid Direct/Iterative Solver for CFD Problems

Thies

Wubs

2011

2011 IEEE Seventh International Conference on eScience

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Abstract-We discuss the parallel implementation of a hybrid direct/iterative solver for a special class of saddle point matrices arising from the discretization of the steady Navier-Stokes equations on an Arakawa C-grid, the F -matrices.The two-level method described here has the following properties: (i) it is very robust, even hat comparatively high Reynolds Numbers; (ii) a single parameter controls fill and convergence, making the method straightforward to use; (iii) the convergence rate is independent of the number of unknowns; (iv) it can be implemented on distributed memory machines in a natural way; (v) the matrix on the second level has the same structure and numerical properties as the original problem, so the method can be applied recursively. The implementation focusses on generality, modularity, code reuse and recursiveness. The solver is implemented using building blocks of the Trilinos libraries. We show its performance on a parallel computer for the NavierStokes equations.

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A New Petrov–Galerkin Smoothed Aggregation Preconditioner for Nonsymmetric Linear Systems

Cited by 68 publications

References 11 publications

Implementation and Scalability Analysis of Balancing Domain Decomposition Methods

Implementation and Scalability Analysis of Balancing Domain Decomposition Methods

Smoothed aggregation for Helmholtz problems

Design of a Parallel Hybrid Direct/Iterative Solver for CFD Problems

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