Improving Multifrontal Methods by Means of Block Low-Rank Representations

Amestoy, Patrick; Ashcraft, Cleve; Boiteau, Olivier; Buttari, Alfredo; L’Excellent, Jean-Yves; Weisbecker, Clément

doi:10.1137/120903476

Cited by 158 publications

(175 citation statements)

References 30 publications

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“…The Intel MKL 2017 is used for BLAS and SVD kernels. The RRQR kernel is issued from the BLR-MUMPS solver [20], and is an extension of the block rank-revealing QR factorization subroutines from LAPACK 3.6.0 (xGEQP3) to stop the factorization when the precision is reached.…”

Section: Methodsmentioning

confidence: 99%

“…Block Low-Rank compression has been investigated for dense matrices [18,19], and for sparse linear systems when using a multifrontal method [20,21]. Considering that these approaches are similar to the current study, a detailed comparison will be described in Section 6.…”

Section: Introductionmentioning

confidence: 92%

“…A traveler salesman strategy is implemented in PaStiX [27] and divides by more than two the number of these off-diagonal blocks. Other approaches [11,20] perform a k-way ordering of supernodes, starting from a reconnected graph of a separator, to order consecutively vertices belonging to a same local part of the separator's graph. Such re-ordering technique also allows to reduce ranks of the low-rank blocks as shown in [20].…”

Section: Background On Sparse Linear Algebramentioning

confidence: 99%

“…Other approaches [11,20] perform a k-way ordering of supernodes, starting from a reconnected graph of a separator, to order consecutively vertices belonging to a same local part of the separator's graph. Such re-ordering technique also allows to reduce ranks of the low-rank blocks as shown in [20]. To introduce more parallelism and data locality, the final structure can then be split into tiles as it is now commonly done in dense linear algebra libraries to fit the computational units granularity.…”

Section: Background On Sparse Linear Algebramentioning

confidence: 99%

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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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Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 92%

Section: Background On Sparse Linear Algebramentioning

confidence: 99%

Section: Background On Sparse Linear Algebramentioning

confidence: 99%

See 2 more Smart Citations

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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“…To solve the resulting linear system we use the MUMPS direct factorisation solver [44] interfaced from PETSc [42].…”

Section: Tangent Linear Modelmentioning

confidence: 99%

Accelerating Monte Carlo estimation with derivatives of high-level finite element models

Hauseux

Hale

Bordas

2017

Computer Methods in Applied Mechanics and Engineering

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In this paper we demonstrate the ability of a derivative-driven Monte Carlo estimator to accelerate the propagation of uncertainty through two high-level non-linear finite element models. The use of derivative information amounts to a correction to the standard Monte Carlo estimation procedure that reduces the variance under certain conditions. We express the finite element models in variational form using the high-level Unified Form Language (UFL). We derive the tangent linear model automatically from this high-level description and use it to efficiently calculate the required derivative information. To study the effectiveness of the derivative-driven method we consider two stochastic PDEs; a onedimensional Burgers equation with stochastic viscosity and a three-dimensional geometrically non-linear Mooney-Rivlin hyperelastic equation with stochastic density and volumetric material parameter. Our results show that for these problems the first-order derivative-driven Monte Carlo method is around one order of magnitude faster than the standard Monte Carlo method and at the cost of only one extra tangent linear solution per estimation problem. We find similar trends when comparing with a modern non-intrusive multi-level polynomial chaos expansion method. We parallelise the task of the repeated forward model evaluations across a cluster using the ipyparallel and mpi4py software tools. A complete working example showing the solution of the stochastic viscous Burgers equation is included as supplementary material.

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Hierarchical Interpolative Factorization for Elliptic Operators: Differential Equations

Ying

2015

Comm Pure Appl Math

133

114

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This paper introduces the hierarchical interpolative factorization for elliptic partial differential equations (HIF‐DE) in two (2D) and three dimensions (3D). This factorization takes the form of an approximate generalized LU/LDL decomposition that facilitates the efficient inversion of the discretized operator. HIF‐DE is based on the nested dissection multifrontal method but uses skeletonization on the separator fronts to sparsify the dense frontal matrices and thus reduce the cost. We conjecture that this strategy yields linear complexity in 2D and quasilinear complexity in 3D. Estimated linear complexity in 3D can be achieved by skeletonizing the compressed fronts themselves, which amounts geometrically to a recursive dimensional reduction scheme. Numerical experiments support our claims and further demonstrate the performance of our algorithm as a fast direct solver and preconditioner. MATLAB® codes are freely available.© 2016 Wiley Periodicals, Inc.

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Improving Multifrontal Methods by Means of Block Low-Rank Representations

Cited by 158 publications

References 30 publications

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

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

Accelerating Monte Carlo estimation with derivatives of high-level finite element models

Hierarchical Interpolative Factorization for Elliptic Operators: Differential Equations

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