Fast Hierarchical Solvers For Sparse Matrices Using Extended Sparsification and Low-Rank Approximation

Pouransari, Hadi; Coulier, Pieter; Darve, Eric

doi:10.1137/15m1046939

Cited by 39 publications

(84 citation statements)

References 55 publications

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“…In comparison to the quadratic complexity of the full-rank solver, most sparse solvers based on hierarchical formats have been shown to possess near-linear complexity. To cite a few, [40,39,22] are HSS-based, [24] is HBS-based, [7] is HODLR-based, and [34] is H 2 -based. Other related work includes [28], a multifrontal solver with front skeletonization, and [37], a Cholesky solver with fill-in compression.…”

mentioning

confidence: 99%

On the Complexity of the Block Low-Rank Multifrontal Factorization

Amestoy¹,

Buttari²,

L’Excellent³

et al. 2017

SIAM J. Sci. Comput.

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Abstract. Matrices coming from elliptic Partial Differential Equations have been shown to have a lowrank property: well defined off-diagonal blocks of their Schur complements can be approximated by low-rank products and this property can be efficiently exploited in multifrontal solvers to provide a substantial reduction of their complexity. Among the possible low-rank formats, the Block Low-Rank format (BLR) is easy to use in a general purpose multifrontal solver and has been shown to provide significant gains compared to full-rank on practical applications. However, unlike hierarchical formats, such as H and HSS, its theoretical complexity was unknown. In this paper, we extend the theoretical work done on hierarchical matrices in order to compute the theoretical complexity of the BLR multifrontal factorization. We then present several variants of the BLR multifrontal factorization, depending on the strategies used to perform the updates in the frontal matrices and on the constraints on how numerical pivoting is handled. We show how these variants can further reduce the complexity of the factorization. In the best case (3D, constant ranks), we obtain a complexity of the order of O(n 4/3 ). We provide an experimental study with numerical results to support our complexity bounds.

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mentioning

confidence: 99%

On the Complexity of the Block Low-Rank Multifrontal Factorization

Amestoy¹,

Buttari²,

L’Excellent³

et al. 2017

SIAM J. Sci. Comput.

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“…In [17], an H 2 sparse algorithm was described. It is similar in many respects to [14], and extends the work of [16]. All these solvers have a guaranteed linear complexity, for a given error tolerance, and assuming a bounded rank for all well-separated pairs of clusters (the admissibility criterion in Hackbusch et al's terminology).…”

Section: Introductionmentioning

confidence: 96%

“…H 2 arithmetic [13] have been used in several sparse solvers. In [14], a fast sparse H 2 solver, called LoRaSp, based on extended sparsification was introduced. In [15], a variant of LoRaSp, aiming at improving the quality of the solver when used as a preconditioner, was presented, as well as a numerical analysis of the convergence with H 2 preconditioning.…”

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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“…More recently, Pouransari et al extended the approaches in Darve et al to solving sparse linear systems, assuming that the fill‐ins during a block Gaussian elimination have low numerical rank for well‐separated interactions. An approximate factorization has been introduced via extended sparsification, which has time and memory complexity of

scriptO false(n \log^{2} 1 false/ ϵ false)

and

scriptO false(n \log 1 false/ ϵ false)

, respectively. Here, ϵ is the accuracy of the low‐rank approximations.…”

Section: Introductionmentioning

confidence: 99%

“…Despite the high accuracy of approximate factorizations, the robustness of this type of approach is subject to the condition number of the original problem. In Pouransari et al, rather than solving directly, the following equation is solved instead:

A_{scriptH} x_{scriptH} = b with false‖ A_{scriptH} - A false‖ \leq ϵ false‖ A false‖ .

Then, the accuracy of the solution is subject to both ϵ and κ ( A )=‖ A ‖‖ A −1 ‖

\frac{‖ x - x_{H} ‖}{‖ x ‖} \leq C ϵ κ (A),

where C is a constant. As the upper bound for the relative accuracy is proportional to the matrix condition number, the accuracy deteriorates when the condition number of matrix grows.…”

Section: Introductionmentioning

confidence: 99%

Sparse hierarchical solvers with guaranteed convergence

Yang

Pouransari

Darve

2019

Numerical Meth Engineering

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Summary Solving sparse linear systems from discretized partial differential equations is challenging. Direct solvers have, in many cases, quadratic complexity (depending on geometry), while iterative solvers require problem dependent preconditioners to be robust and efficient. Approximate factorization preconditioners such as incomplete LU factorization provide cheap approximations to the system matrix. However, even a highly accurate preconditioner may have deteriorating performance when the condition number of the system matrix increases. By increasing the accuracy on low‐frequency errors, we propose a novel hierarchical solver with improved robustness with respect to the condition number of the linear system. This solver retains the linear computational cost and memory footprint of the original algorithm.

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Fast Hierarchical Solvers For Sparse Matrices Using Extended Sparsification and Low-Rank Approximation

Cited by 39 publications

References 55 publications

On the Complexity of the Block Low-Rank Multifrontal Factorization

On the Complexity of the Block Low-Rank Multifrontal Factorization

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

Sparse hierarchical solvers with guaranteed convergence

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