Performance Of H-Lu Preconditioning For Sparse Matrices

Grasedyck, Lars; Hackbusch, Wolfgang; Kriemann, Ronald

doi:10.2478/cmam-2008-0024

Cited by 23 publications

(25 citation statements)

References 26 publications

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“…Again, numerical evidence indicates their great usefulness for example, for black box preconditioning in iterative solvers, [Beb05,Gra05,GHK08,LBG06,GKLB08].…”

Section: Introductionmentioning

confidence: 99%

Existence of $\mathcal {H}$-matrix approximants to the inverses of BEM matrices: The simple-layer operator

2015

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Abstract. We consider the question of approximating the inverse W = V −1 of the Galerkin stiffness matrix V obtained by discretizing the simple-layer operator V with piecewise constant functions. The block partitioning of W is assumed to satisfy any one of several standard admissibility criteria that are employed in connection with clustering algorithms to approximate the discrete BEM operator V. We show that W can be approximated by blockwise low-rank matrices such that the error decays exponentially in the block rank employed. Similar exponential approximability results are shown for the Cholesky factorization of V.

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“…Again, numerical evidence indicates their great usefulness for example, for black box preconditioning in iterative solvers, [Beb05,Gra05,GHK08,LBG06,GKLB08].…”

Section: Introductionmentioning

confidence: 99%

Existence of $\mathcal {H}$-matrix approximants to the inverses of BEM matrices: The simple-layer operator

2015

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“…Performance, as well as a comparison of H-LU with some sparse direct solvers is presented in [3]. Kriemann [4] and Lizé [5] implemented this algorithm using Directed Acyclic Graphs.…”

Section: Introductionmentioning

confidence: 99%

Sparse supernodal solver using block low-rank compression: Design, performance and analysis

Pichon

Darve

Faverge

et al. 2018

Journal of Computational Science

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Abstract:This paper presents two approaches using a Block Low-Rank (BLR) compression technique to reduce the memory footprint and/or the time-to-solution of the sparse supernodal solver PaStiX. This flat, non-hierarchical, compression method allows to take advantage of the low-rank property of the blocks appearing during the factorization of sparse linear systems, which come from the discretization of partial differential equations. The first approach, called Minimal Memory, illustrates the maximum memory gain that can be obtained with the BLR compression method, while the second approach, called Just-In-Time, mainly focuses on reducing the computational complexity and thus the time-to-solution. Singular Value Decomposition (SVD) and Rank-Revealing QR (RRQR), as compression kernels, are both compared in terms of factorization time, memory consumption, as well as numerical properties. Experiments on a single node with 24 threads and 128 GB of memory are performed to evaluate the potential of both strategies. On a set of matrices from real-life problems, we demonstrate a memory footprint reduction of up to 4 times using the Minimal Memory strategy and a computational time speedup of up to 3.5 times with the Just-In-Time strategy. Then, we study the impact of configuration parameters of the BLR solver that allowed us to solve a 3D laplacian of 36 million unknowns a single node, while the full-rank solver stopped at 8 million due to memory limitation. Key-words:Sparse linear solver, block low-rank compression, PaStiX direct solver, multithreaded architectures * Inria Bordeaux -Sud-Ouest, Talence, France † University of Bordeaux, Talence, France ‡ CNRS (Labri UMR 5800), Talence, France § Mechanical Engineering Department, Stanford University, United States ¶ Bordeaux INP, Talence, France Un solveur supernodal creux utilisant une compression de rang faible par bloc: conception, étude de performance et analyse des résultats Résumé :Ce papier présente deux approches utilisant les techniques de compression de rang faible par bloc (BLR) afin de réduire l'empreinte mémoire et/ou le temps de résolution du solveur superndal PaStiX. Cette technique de compression à plat, non hiérarchique, permet de tirer parti des propriétés de rang faible dans les blocs obtenus lors de la factorisation du système linéaire provenant par exemple de la discrétisation des équations différentielles partielles. La première approche, appelée Minimal Memory, montre le gain mémoire maximum qu'il est possible d'obtenir avec une compression BLR, alors que la seconde approche, appelée Just-InTime, se concentre principalement sur la réduction du temps de calcul pour la résolution du système. Dans cette étude, nous comparons, en termes de temps de calcul et de consommation mémoire, les noyaux de compression qui utilisent soit la technique de décomposition en valeurs propres singulières (SVD) soit la factorisation QR avec détermination du rang (RRQR). Nous avons réalisé les expériences sur un noeud composé de 24 coeurs avec 128 GB de mémoire sur une collec...

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“…An exception is the work by Grasedyck et al [32], where their H-LU preconditioner is compared with BoomerAMG, Pardiso, MUMPS, UMFPACK, SuperLU, and Spooles. However, their executions are serial, and show that their H-matrix code is not yet competitive with these other highly optimized libraries.…”

Section: Introductionmentioning

confidence: 99%

Fast multipole preconditioners for sparse matrices arising from elliptic equations

Ibeid

Yokota

Pestana

et al. 2017

Comput. Visual Sci.

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Among optimal hierarchical algorithms for the computational solution of elliptic problems, the fast multipole method (FMM) stands out for its adaptability to emerging architectures, having high arithmetic intensity, tunable accuracy, and relaxable global synchronization requirements. We demonstrate that, beyond its traditional use as a solver in problems for which explicit free-space kernel representations are available, the FMM has applicability as a preconditioner in finite domain elliptic boundary value problems, by equipping it with boundary integral capability for satisfying conditions at finite boundaries and by wrapping it in a Krylov method for extensibility to more general operators. Here, we do not discuss the well developed applications of FMM to implement matrix-vector multiplications within Krylov solvers of boundary element methods. Instead, we propose using FMM for the volume-to-volume contribution of inhomogeneous Poisson-like problems, where the boundary integral is a small part of the overall computation. Our King Abdullah University of Science and Technology, Thuwal, Saudi Arabia method may be used to precondition sparse matrices arising from finite difference/element discretizations, and can handle a broader range of scientific applications. It is capable of algebraic convergence rates down to the truncation error of the discretized PDE comparable to those of multigrid methods, and it offers potentially superior multicore and distributed memory scalability properties on commodity architecture supercomputers. Compared with other methods exploiting the low-rank character of off-diagonal blocks of the dense resolvent operator, FMM-preconditioned Krylov iteration may reduce the amount of communication because it is matrix-free and exploits the tree structure of FMM. We describe our tests in reproducible detail with freely available codes and outline directions for further extensibility.

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Performance Of H-Lu Preconditioning For Sparse Matrices

Cited by 23 publications

References 26 publications

Existence of $\mathcal {H}$-matrix approximants to the inverses of BEM matrices: The simple-layer operator

Existence of $\mathcal {H}$-matrix approximants to the inverses of BEM matrices: The simple-layer operator

Sparse supernodal solver using block low-rank compression: Design, performance and analysis

Fast multipole preconditioners for sparse matrices arising from elliptic equations

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