Parallel computation of entries of A-1

In this paper, we are concerned about computing in parallel several entries of the inverse of a large sparse matrix. We assume that the matrix has already been factorized by a direct method and that the factors are distributed. Entries are efficiently computed by exploiting sparsity of the right-hand sides and the solution vectors in the triangular solution phase. We demonstrate that in this setting, parallelism and computational efficiency are two contrasting objectives. We develop an efficient approach and show its efficacy by runs using the MUMPS code that implements a parallel multifrontal method

Modeling 1D distributed-memory dense kernels for an asynchronous multifrontal sparse solver

To solve sparse systems of linear equations, multifrontal methods rely on dense partial LU decompositions of so-called frontal matrices; we consider a parallel asynchronous setting in which several frontal matrices can be factored simultaneously. In this context, to address performance and scalability issues of acyclic pipelined asynchronous factorization kernels, we study models to revisit properties of left and right-looking variants of partial $LU$ decompositions, study the use of several levels of blocking, before focusing on communication issues. The general purpose sparse solver MUMPS has been modified to implement the proposed algorithms and confirm the properties demonstrated by the models

Hybrid scheduling for the parallel solution of linear systems

In this paper, we consider the problem of designing a dynamic scheduling strategy that takes into account both workload and memory information in the context of the parallel multifrontal factorization. The originality of our approach is that we base our estimations (work and memory) on a static optimistic scenario during the analysis phase. This scenario is then used during the factorization phase to constrain the dynamic decisions. The task scheduler has been redesigned to take into account these new features. Moreover performance have been improved because the new constraints allow the new scheduler to make optimal decisions that were forbidden or too dangerous in unconstrained formulations. Performance analysis show that the memory estimation becomes much closer to the memory effectively used and that even in a constrained memory environment we decrease the factorization time with respect to the initial approach.Nous proposons des stratégies d'ordonnancement bi-critères, qui s'intéressent à la fois à la performance et à la consommation mémoire d'un algorithme parallèle de factorisation de matrices creuses, basé sur la méthode multifrontale. L'originalité de notre approche est que nous basons nos estimations mémoire sur un scénario optimiste (simulation lors de la phase d'analyse),qui est ensuite utilisé lors de la factorisation pour contraindre les décisions dynamiques d'ordonnancement. Un nouvel ordonnanceur a été implanté, qui prend en compte ces nouvelles contraintes. De plus, la performance a été améliorée parce que notre nouvelle approche permet à l'ordonnanceur de prendre des décisions meilleures, qui étaient interdites ou trop dangereuses auparavant. Une analyse de performance montre que les estimations mémoire sont beaucoup plus proches de la mémoire effectivement utilisée, et que le temps de factorisation est amélioré de façon significative par rapport à l'approche initiale

Grouping variables in Frontal Matrices to improve Low-Rank Approximations in a Multifrontal Solver

Session 3International audienc

Improving multifrontal methods by means of block low-rank representations

Submitted for publication to SIAMMatrices coming from elliptic Partial Differential Equations (PDEs) have been shown to have a low-rank property: well defined off-diagonal blocks of their Schur complements can be approximated by low-rank products. Given a suitable ordering of the matrix which gives to the blocks a geometrical meaning, 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 gives efficient ways to perform many of the basic dense algebra operations. Several strategies have been proposed to exploit this property. We propose a low-rank format called Block Low-Rank (BLR), and explain how it can be used to reduce the memory footprint and the complexity of direct solvers for sparse matrices based on the multifrontal method. We present experimental results that show how the BLR format delivers gains that are comparable to those obtained with hierarchical formats such as Hierarchical matrices (H matrices) and Hierarchically Semi-Separable (HSS matrices) but provides much greater flexibility and ease of use which are essential in the context of a general purpose, algebraic solver

Robust memory-aware mappings for parallel multifrontal factorizations

International audienceWe study the memory scalability of the parallel multifrontal factorization of sparse matrices. In particular, we are interested in controlling the active memory specific to the multifrontal factorization. We illustrate why commonly used mapping strategies (e.g., the proportional mapping) cannot provide a high memory efficiency, which means that they tend to let the memory usage of the factorization grow when the number of processes increases. We propose “memory-aware” algorithms that aim at maximizing the granularity of parallelism while respecting memory constraints. These algorithms provide accurate memory estimates prior to the factorization and can significantly enhance the robustness of a multifrontal code. We illustrate our approach with experiments performed on large matrices

Parallel computation of entries in A-1

International audienceIn this paper, we consider the computation in parallel of several entries of the inverseof a large sparse matrix. We assume that the matrix has already been factorized by a direct methodand that the factors are distributed. Entries are efficiently computed by exploiting sparsity of theright-hand sides and the solution vectors in the triangular solution phase. We demonstrate that inthis setting, parallelism and computational efficiency are two contrasting objectives. We develop anefficient approach and show its efficiency on a general purpose parallel multifrontal solver

Shared memory parallelism and low-rank approximation techniques applied to direct solvers in FEM simulation

International audienceIn this paper, the performance of a parallel sparse direct solver on a shared memory multicore system is presented. Large size test matrices arising from finite element simulation of induction heating industrial applications are used in order to evaluate the performance improvements due to low-rank representations and multicore parallelizatio

Recent advances in sparse direct solvers

International audienceDirect methods for the solution of sparse systems of linear equations of the form A x = b are used in a wide range of numerical simulation applications. Such methods are based on the decomposition of the matrix into a product of triangular factors (e.g., A = L U ), followed by triangular solves. They are known for their numerical accuracy and robustness but are also characterized by a high memory consumption and a large amount of computations. Here we survey some research directions that are being investigated by the sparse direct solver community to alleviate these issues: memory-aware scheduling techniques, low-rank approximations, and distributed/shared memory hybrid programming

Contributions à la recherche en calcul scientifique haute performance pour les matrices creuses

Nous nous intéressons au développement d'un nouvel algorithme pour estimer la norme d'une matrice de manière incrémentale, à l'implantation d'un modèle de référence des Basic Linear Algebra Subprograms for sparse matrices (Sparse BLAS), et à la réalisation d'un nouveau gestionnaire de tâches pour MUMPS, un solveur multifrontal pour des architectures à mémoire distribuée. Notre méthode pour estimer la norme d'une matrice s'applique aux matrices denses et creuses. Elle peut s'avérer utile dans le cadre des factorisations QR, Cholesky, ou LU. Le standard Sparse BLAS définit des interfaces génériques. Nous avons été amenés à répondre aux questions concernant la représentation et la gestion des données. Le séquencement de tâches devient un enjeu important dès que nous travaillons sur un grand nombre de processeurs. Grâce à notre nouvelle approche, nous pouvons améliorer le passage a l'échelle du solveur MUMPS.TOULOUSE-ENSEEIHT (315552331) / SudocSudocFranceF