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

Abstract

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...

Solving block low-rank linear systems by LU factorization is numerically stable

“…Even though the BLR format has been extensively studied and widely used in numerous applications [1,2,3,4,5,7,12,23,25,27,28,30,31,33], little is known about its numerical behavior in floating-point arithmetic. Indeed, no rounding error analysis has been published for BLR matrix algorithms.…”

“…Another widely used algorithm is the CUF variant (see, for example, [4,28]), which compresses the entire matrix A and then computes its BLR LU factorization. The CUF algorithm is very similar to the CFU one, only differing in that, at line 15 of Algorithm 4.2, the blocks A ik (and A ki ) are already in LR form.…”

Improving the Complexity of Block Low-Rank Factorizations with Fast Matrix Arithmetic

“…Even though the BLR format achieves a higher theoretical complexity than hierarchical formats, its simplicity and flexibility make it easy to use in the context of a general purpose, algebraic solver [6,33,4,36]. Due to its non-hierarchical nature, the BLR format is particularly efficient on parallel computers [6,35,14,1].…”

Analysis of Parallelization Strategies in the context of Hierarchical Matrix Factorizations

“…• PaStiX BLR [102,103] provides BLR compression techniques to compute the sparse supernodal solver PaStiX, either as a direct solver operating at a lower precision or as a preconditioner. The implementations included in PaStiX BLR can be executed in shared memory parallel platforms.…”

“…On the other hand, in general, the definition and storage of H-Matrices leads to complex data accesses. This fact promoted the appearance of alternative structures, such as BLR [10,89,103] and lattice H-Matrices [68,116], that trade off slightly higher time and memory costs in exchange for superior simplicity. One asset of these approaches it that they make it easier to exploit parallelism, as they present more regular structures.…”