IMF: An Incomplete Multifrontal $LU$-Factorization for Element-Structured Sparse Linear Systems

Vannieuwenhoven, Nick; Meerbergen, Karl

doi:10.1137/100818996

Cited by 11 publications

(8 citation statements)

References 55 publications

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“…The results of this chapter show that, by taking advantage of this block structure, the solver can be more robust and efficient. Other recent studies on block ILU preconditioners have drawn similar conclusions on the importance of exposing dense blocks during the construction of the incomplete LU factorization for better performance, in the design of incomplete multifrontal LUfactorization preconditioners [61] and adaptive blocking approaches for blocked incomplete Cholesky factorization [62]. We believe that the proposed VBARMS method can be useful for solving linear systems also in other areas, such as in Electromagnetics applications [63][64][65].…”

Section: Discussionsupporting

confidence: 61%

Multilevel Variable-Block Schur-Complement-Based Preconditioning for the Implicit Solution of the Reynolds- Averaged Navier-Stokes Equations Using Unstructured Grids

Carpentieri¹,

Bonfiglioli²

2018

Computational Fluid Dynamics - Basic Instruments and Applications in Science

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Implicit methods based on the Newton's rootfinding algorithm are receiving an increasing attention for the solution of complex Computational Fluid Dynamics (CFD) applications due to their potential to converge in a very small number of iterations. This approach requires fast convergence acceleration techniques in order to compete with other conventional solvers, such as those based on artificial dissipation or upwind schemes, in terms of CPU time. In this chapter, we describe a multilevel variable-block Schur-complement-based preconditioning for the implicit solution of the Reynoldsaveraged Navier-Stokes equations using unstructured grids on distributed-memory parallel computers. The proposed solver detects automatically exact or approximate dense structures in the linear system arising from the discretization, and exploits this information to enhance the robustness and improve the scalability of the block factorization. A complete study of the numerical and parallel performance of the solver is presented for the analysis of turbulent Navier-Stokes equations on a suite of threedimensional test cases.

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Section: Discussionsupporting

confidence: 61%

Multilevel Variable-Block Schur-Complement-Based Preconditioning for the Implicit Solution of the Reynolds- Averaged Navier-Stokes Equations Using Unstructured Grids

Carpentieri¹,

Bonfiglioli²

2018

Computational Fluid Dynamics - Basic Instruments and Applications in Science

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“…This approach is nothing but a block form of Gaussian elimination and it is generally costly although there are practical alternatives discussed in the literature [11,21,82,118] that are based on Schur complements. However, it is worth pointing out that, viewed from this angle, the goal of all AMG methods is essentially to find inexpensive approximations to the Schur complement system.…”

Section: Algebraic Multigridmentioning

confidence: 99%

“…For AMG, the graph representation of the problem at a certain level is explicitly 'coarsened' by using various mechanisms [28,101,102]. Since these mechanisms are geared toward a certain class of problems, essentially originating from Poisson-like partial differential equations, researchers later sought to extend AMG ideas in order to define algebraic techniques based on incomplete LU (ILU) factorizations [4][5][6][7][8][9]11,23,90,118].…”

Section: Introductionmentioning

confidence: 99%

Graph coarsening: from scientific computing to machine learning

Chen

Saad

Zhang

2022

SeMA

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The general method of graph coarsening or graph reduction has been a remarkably useful and ubiquitous tool in scientific computing and it is now just starting to have a similar impact in machine learning. The goal of this paper is to take a broad look into coarsening techniques that have been successfully deployed in scientific computing and see how similar principles are finding their way in more recent applications related to machine learning. In scientific computing, coarsening plays a central role in algebraic multigrid methods as well as the related class of multilevel incomplete LU factorizations. In machine learning, graph coarsening goes under various names, e.g., graph downsampling or graph reduction. Its goal in most cases is to replace some original graph by one which has fewer nodes, but whose structure and characteristics are similar to those of the original graph. As will be seen, a common strategy in these methods is to rely on spectral properties to define the coarse graph.

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“…Dense blocks have been used successfully in the past [6,7,18,29,31,34,35,36,39,45,48,55] to enhance the performance of incomplete factorization preconditioners. Many of these block algorithms are designed for symmetric positive definite (SPD) systems, which we covered in an earlier publication [29].…”

Section: Related Work LI Saad and Chowmentioning

confidence: 99%

Enhancing Performance and Robustness of ILU Preconditioners by Blocking and Selective Transposition

Gupta¹

2017

SIAM J. Sci. Comput.

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Incomplete factorization is one of the most effective general-purpose preconditioning methods for Krylov subspace solvers for large sparse systems of linear equations. Two techniques for enhancing the robustness and performance of incomplete LU factorization for sparse unsymmetric systems are described. A block incomplete factorization algorithm based on the Crout variation of LU factorization is presented. The algorithm is suitable for incorporating threshold-based dropping as well as unrestricted partial pivoting, and it overcomes several limitations of existing incomplete LU factorization algorithms with and without blocking. It is shown that blocking has a threepronged impact: it speeds up the computation of incomplete factors and the solution of the associated triangular systems, it permits denser and more robust factors to be computed economically, and it permits a trade-off with the restart parameter of GMRES to further improve the overall speed and robustness. A highly effective heuristic for improving the quality of preconditioning and subsequent convergence of the associated iterative method is presented. The choice of the Crout variant as the underlying factorization algorithm enables efficient implementation of this heuristic, which has the potential to improve both incomplete and complete sparse LU factorization of matrices that require pivoting for numerical stability.

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IMF: An Incomplete Multifrontal $LU$-Factorization for Element-Structured Sparse Linear Systems

Cited by 11 publications

References 55 publications

Multilevel Variable-Block Schur-Complement-Based Preconditioning for the Implicit Solution of the Reynolds- Averaged Navier-Stokes Equations Using Unstructured Grids

Multilevel Variable-Block Schur-Complement-Based Preconditioning for the Implicit Solution of the Reynolds- Averaged Navier-Stokes Equations Using Unstructured Grids

Graph coarsening: from scientific computing to machine learning

Enhancing Performance and Robustness of ILU Preconditioners by Blocking and Selective Transposition

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