Sympiler: Transforming Sparse Matrix Codes by Decoupling Symbolic Analysis

Sympiler is a domain-specific code generator that optimizes sparse matrix computations by decoupling the symbolic analysis phase from the numerical manipulation stage in sparse codes. The computation patterns in sparse numerical methods are guided by the input sparsity structure and the sparse algorithm itself. In many real-world simulations, the sparsity pattern changes little or not at all. Sympiler takes advantage of these properties to symbolically analyze sparse codes at compile-time and to apply inspector-guided transformations that enable applying low-level transformations to sparse codes. As a result, the Sympiler-generated code outperforms highly-optimized matrix factorization codes from commonly-used specialized libraries, obtaining average speedups over Eigen and CHOLMOD of 3.8X and 1.5X respectively.Comment: 12 page

Combinatorial problems in solving linear systems

42 pages, available as LIP research report RR-2009-15Numerical linear algebra and combinatorial optimization are vast subjects; as is their interaction. In virtually all cases there should be a notion of sparsity for a combinatorial problem to arise. Sparse matrices therefore form the basis of the interaction of these two seemingly disparate subjects. As the core of many of today's numerical linear algebra computations consists of the solution of sparse linear system by direct or iterative methods, we survey some combinatorial problems, ideas, and algorithms relating to these computations. On the direct methods side, we discuss issues such as matrix ordering; bipartite matching and matrix scaling for better pivoting; task assignment and scheduling for parallel multifrontal solvers. On the iterative method side, we discuss preconditioning techniques including incomplete factorization preconditioners, support graph preconditioners, and algebraic multigrid. In a separate part, we discuss the block triangular form of sparse matrices

Scilab and MATLAB Interfaces to MUMPS (version 4.6 or greater)

This document describes the Scilab and MATLAB interfaces to MUMPS version 4.6. We describe the differences and similarities between usual Fortran/C MUMPS interfaces and its Scilab/MATLAB interfaces, the calling sequences and functionalities. Examples of use and experimental results are also provided.Nous décrivons les séquences d’appel et les fonctionnalités de nos interfaces Scilab/MATLAB et nous évoquons ses différences et similarités avec les interfaces Fortran/C habituelles de MUMPS. Nous présentons aussi des exemples d’utilisation et quelques résultats expérimentau

A new numerical method to construct binary neutron star initial data

We present a new numerical method for the generation of binary neutron star initial data using a method along the lines of the the Wilson-Mathews or the closely related conformal thin sandwich approach. Our method uses six different computational domains, which include spatial infinity. Each domain has its own coordinates which are chosen such that the star surfaces always coincide with domain boundaries. These properties facilitate the imposition of boundary conditions. Since all our fields are smooth inside each domain, we are able to use an efficient pseudospectral method to solve the elliptic equations associated with the conformal thin sandwich approach. Currently we have implemented corotating configurations with arbitrary mass ratios, but an extension to arbitrary spins is possible. The main purpose of this paper is to introduce our new method and to test our code for several different configurations.Comment: 18 pages, 8 figures, 1 tabl

Résolution triangulaire de systèmes linéaires creux de grande taille dans un contexte parallèle multifrontal et hors-mémoire

Nous nous intéressons à la résolution de systèmes linéaires creux de très grande taille par des méthodes directes de factorisation. Dans ce contexte, la taille de la matrice des facteurs constitue un des facteurs limitants principaux pour l'utilisation de méthodes directes de résolution. Nous supposons donc que la matrice des facteurs est de trop grande taille pour être rangée dans la mémoire principale du multiprocesseur et qu'elle a donc été écrite sur les disques locaux (hors-mémoire : OOC) d'une machine multiprocesseurs durant l'étape de factorisation. Nous nous intéressons à l'étude et au développement de techniques efficaces pour la phase de résolution après une factorization multifrontale creuse. La phase de résolution, souvent négligée dans les travaux sur les méthodes directes de résolution directe creuse, constitue alors un point critique de la performance de nombreuses applications scientifiques, souvent même plus critique que l'étape de factorisation. Cette thèse se compose de deux parties. Dans la première partie nous nous proposons des algorithmes pour améliorer la performance de la résolution hors-mémoire. Dans la deuxième partie nous pousuivons ce travail en montrant comment exploiter la nature creuse des seconds membres pour réduire le volume de données accédées en mémoire. Dans la première partie de cette thèse nous introduisons deux approches de lecture des données sur le disque dur. Nous montrons ensuite que dans un environnement parallèle le séquencement des tâches peut fortement influencer la performance. Nous prouvons qu'un ordonnancement contraint des tâches peut être introduit; qu'il n'introduit pas d'interblocage entre processus et qu'il permet d'améliorer les performances. Nous conduisons nos expériences sur des problèmes industriels de grande taille (plus de 8 Millions d'inconnues) et utilisons une version hors-mémoire d'un code multifrontal creux appelé MUMPS (solveur multifrontal parallèle). Dans la deuxième partie de ce travail nous nous intéressons au cas de seconds membres creux multiples. Ce problème apparaît dans des applications en electromagnétisme et en assimilation de données et résulte du besoin de calculer l'espace propre d'une matrice fortement déficiente, du calcul d'éléments de l'inverse de la matrice associée aux équations normales pour les moindres carrés linéaires ou encore du traitement de matrices fortement réductibles en programmation linéaire. Nous décrivons un algorithme efficace de réduction du volume d'Entrées/Sorties sur le disque lors d'une résolution hors-mémoire. Plus généralement nous montrons comment le caractère creux des seconds -membres peut être exploité pour réduire le nombre d'opérations et le nombre d'accès à la mémoire lors de l'étape de résolution. Le travail présenté dans cette thèse a été partiellement financé par le projet SOLSTICE de l'ANR (ANR-06-CIS6-010). ABSTRACT : We consider the solution of very large systems of linear equations with direct multifrontal methods. In this context the size of the factors is an important limitation for the use of sparse direct solvers. We will thus assume that the factors have been written on the local disks of our target multiprocessor machine during parallel factorization. Our main focus is the study and the design of efficient approaches for the forward and backward substitution phases after a sparse multifrontal factorization. These phases involve sparse triangular solution and have often been neglected in previous works on sparse direct factorization. In many applications, however, the time for the solution can be the main bottleneck for the performance. This thesis consists of two parts. The focus of the first part is on optimizing the out-of-core performance of the solution phase. The focus of the second part is to further improve the performance by exploiting the sparsity of the right-hand side vectors. In the first part, we describe and compare two approaches to access data from the hard disk. We then show that in a parallel environment the task scheduling can strongly influence the performance. We prove that a constraint ordering of the tasks is possible; it does not introduce any deadlock and it improves the performance. Experiments on large real test problems (more than 8 million unknowns) using an out-of-core version of a sparse multifrontal code called MUMPS (MUltifrontal Massively Parallel Solver) are used to analyse the behaviour of our algorithms. In the second part, we are interested in applications with sparse multiple right-hand sides, particularly those with single nonzero entries. The motivating applications arise in electromagnetism and data assimilation. In such applications, we need either to compute the null space of a highly rank deficient matrix or to compute entries in the inverse of a matrix associated with the normal equations of linear least-squares problems. We cast both of these problems as linear systems with multiple right-hand side vectors, each containing a single nonzero entry. We describe, implement and comment on efficient algorithms to reduce the input-output cost during an outof- core execution. We show how the sparsity of the right-hand side can be exploited to limit both the number of operations and the amount of data accessed. The work presented in this thesis has been partially supported by SOLSTICE ANR project (ANR-06-CIS6-010)

High Performance Computing of Three-Dimensional Finite Element Codes on a 64-bit Machine

Three dimensional Navier-Stokes finite element formulations require huge computational power in terms of memory and CPU time. Recent developments in sparse direct solvers have significantly reduced the memory and computational time of direct solution methods. The objective of this study is twofold. First is to evaluate the performance of various state-of-the-art sequential sparse direct solvers in the context of finite element formulation of fluid flow problems. Second is to examine the merit in upgrading from 32 bit machine to a 64 bit machine with larger RAM capacity in terms of its capacity to solve larger problems. The choice of a direct solver is dependent on its computational time and its in-core memory requirements. Here four different solvers, UMFPACK, MUMPS, HSL_MA78 and PARDISO are compared. The performances of these solvers with respect to the computational time and memory requirements on a 64-bit windows server machine with 16GB RAM is evaluated

Some Experiments and Issues to Exploit Multicore Parallelism in a Distributed-Memory Parallel Sparse Direct Solver

MUMPS is a parallel sparse direct solver, using message passing (MPI) for parallelism. In this report we experiment how thread parallelism can help taking advantage of recent multicore architectures. The work done consists in testing multithreaded BLAS libraries and inserting OpenMP directives in the routines revealed to be costly by profiling, with the objective to avoid any deep restructuring or rewriting of the code. We report on various aspects of this work, present some of the benefits and difficulties, and show that 4 threads per MPI process is generally a good compromise. We then discuss various issues that appear to be critical in a mixed MPI-OpenMP environment

Parallel triangular solution in the out-of-core multifrontal approach for solving large sparse linear systems

We consider the solution of very large systems of linear equations with direct multifrontal methods. In this context the size of the factors is an important limitation for the use of sparse direct solvers. We will thus assume that the factors have been written on the local disks of our target multiprocessor machine during parallel factorization. Our main focus is the study and the design of efficient approaches for the forward and backward substitution phases after a sparse multifrontal factorization. These phases involve sparse triangular solution and have often been neglected in previous works on sparse direct factorization. In many applications, however, the time for the solution can be the main bottleneck for the performance. This thesis consists of two parts. The focus of the first part is on optimizing the out-of-core performance of the solution phase. The focus of the second part is to further improve the performance by exploiting the sparsity of the right-hand side vectors. In the first part, we describe and compare two approaches to access data from the hard disk. We then show that in a parallel environment the task scheduling can strongly influence the performance. We prove that a constraint ordering of the tasks is possible; it does not introduce any deadlock and it improves the performance. Experiments on large real test problems (more than 8 million unknowns) using an out-of-core version of a sparse multifrontal code called MUMPS (MUltifrontal Massively Parallel Solver) are used to analyse the behaviour of our algorithms. In the second part, we are interested in applications with sparse multiple right-hand sides, particularly those with single nonzero entries. The motivating applications arise in electromagnetism and data assimilation. In such applications, we need either to compute the null space of a highly rank deficient matrix or to compute entries in the inverse of a matrix associated with the normal equations of linear least-squares problems. We cast both of these problems as linear systems with multiple right-hand side vectors, each containing a single nonzero entry. We describe, implement and comment on efficient algorithms to reduce the input-output cost during an outof- core execution. We show how the sparsity of the right-hand side can be exploited to limit both the number of operations and the amount of data accessed. The work presented in this thesis has been partially supported by SOLSTICE ANR project (ANR-06-CIS6-010)