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1. Block Low-Rank Matrices with Shared Bases: Potential and Limitations of the BLR^2 Format

Author

Alfredo Buttari, Théo Mary, Cleve Ashcraft, ANSYS, Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées, Performance et Qualité des Algorithmes Numériques (PEQUAN), Laboratoire d'Informatique de Paris 6 (LIP6), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Centre National de la Recherche Scientifique (CNRS), and LIP6

Subjects
Rank (linear algebra), Astrophysics::High Energy Astrophysical Phenomena, 010103 numerical & computational mathematics, Astrophysics::Cosmology and Extragalactic Astrophysics, computer.software_genre, 01 natural sciences, law.invention, Separable space, Matrix (mathematics), law, numerical linear algebra, [INFO]Computer Science [cs], 0101 mathematics, [MATH]Mathematics [math], Column (data store), Astrophysics::Galaxy Astrophysics, Mathematics, Block (data storage), Discrete mathematics, hierarchical matrices, Numerical linear algebra, block low-rank matrices, Special class, LU decomposition, Data sparse matrices, LU factorization, computer, Analysis, block separable matrices
Abstract

International audience; We investigate a special class of data sparse rank-structured matrices that combine a flat block low-rank (BLR) partitioning with the use of shared (called nested in the hierarchical case) bases. This format is to H 2 matrices what BLR is to H matrices: we therefore call it the BLR 2 matrix format. We present algorithms for the construction and LU factorization of BLR 2 matrices, and perform their cost analysis-both asymptotically and for a fixed problem size. With weak admissibility, BLR 2 matrices reduce to block separable matrices (the flat version of HBS/HSS). Our analysis and numerical experiments reveal some limitations of BLR 2 matrices with weak admissibility, which we propose to overcome with two approaches: strong admissibility, and the use of multiple shared bases per row and column.

Published
2020

2. Improving Multifrontal Methods by Means of Block Low-Rank Representations

Author

Patrick R. Amestoy, Olivier Boiteau, Cleve Ashcraft, Jean-Yves L'Excellent, Clement Weisbecker, Alfredo Buttari, Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Livermore Software Technology Corporation [Livermore] (LSTC), EDF R&D (EDF R&D), EDF (EDF), Centre National de la Recherche Scientifique (CNRS), Optimisation des ressources : modèles, algorithmes et ordonnancement (ROMA), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de l'Informatique du Parallélisme (LIP), École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Toulouse Mind & Brain Institut (TMBI), Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), Institut National Polytechnique (Toulouse) (Toulouse INP), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), and Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL)

Subjects
multifrontal method, sparse direct methods, Rank (linear algebra), Applied Mathematics, MathematicsofComputing_NUMERICALANALYSIS, Solver, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, QR decomposition, Algebra, Computational Mathematics, Matrix (mathematics), Algebraic operation, low-rank approximations, Singular value decomposition, [INFO]Computer Science [cs], elliptic, PDEs, Algorithm, [INFO.INFO-MS]Computer Science [cs]/Mathematical Software [cs.MS], Block (data storage), Sparse matrix, Mathematics
Abstract

International audience; Matrices coming from elliptic partial differential equations have been shown to have alow-rank property: well-defined off-diagonal blocks of their Schur complements can be approximatedby low-rank products. Given a suitable ordering of the matrix which gives the blocks a geometricalmeaning, such approximations can be computed using an SVD or a rank-revealing QR factorization.The resulting representation offers a substantial reduction of the memory requirement and givesefficient ways to perform many of the basic dense linear algebra operations. Several strategies,mostly based on hierarchical formats, have been proposed to exploit this property. We study asimple, nonhierarchical, low-rank format called block low-rank (BLR) and explain how it can be usedto reduce the memory footprint and the complexity of sparse direct solvers based on the multifrontalmethod. We present experimental results on matrices coming from elliptic PDEs and from variousother applications. We show that even if BLR-based factorizations are asymptotically less efficientthan hierarchical approaches, they still deliver considerable gains. The BLR format is compatiblewith numerical pivoting, and its simplicity and flexibility make it easy to use in the context of ageneral purpose, algebraic solver.

Published
2015

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