Optimization of Triangular and Banded Matrix Operations Using 2d-Packed Layouts

International audienceOver the past few years, multicore systems have become more and more powerful and, thereby, very useful in high-performance computing. However, many applications, such as some linear algebra algorithms, still cannot take full advantage of these systems. This is mainly due to the shortage of optimization techniques dealing with irregular control structures. In particular, the well-known polyhedral model fails to optimize loop nests whose bounds and/or array references are not affine functions. This is more likely to occur when handling sparse matrices in their packed formats. In this paper, we propose to use 2d-packed layouts and simple affine transformations to enable optimization of triangular and banded matrix operations. The benefit of our proposal is shown through an experimental study over a set of linear algebra benchmarks

Static versus Dynamic Memory Allocation: a Comparison for Linear Algebra Kernels

International audienceThe polyhedral model permits to automatically improve data locality and enable parallelism of regular linear algebra kernels. In previous work we have proposed a new data structure, 2d-packed layout, to store only the non-zeros elements of regular sparse (triangular and banded) matrices dynamically allocated for different basic linear algebra operations, and used Pluto to parallelize and optimize them. To our surprise, there were huge discrepancies in our measures of these kernels execution times that were due to the allocation mode: as statically declared arrays or as dynamically allocated arrays of pointers.In this paper we compare the performance of various linear algebra kernels, including some linear algebra kernels from the PolyBench suite, using different array allocation modes. We present our detailed investigation of the possible reasons of the performance variation on two different architectures: a dual 12-cores AMD (Magny-Cours) and a dual 10-cores Intel Xeon (Haswell-EP).We conclude that static or dynamic memory allocation has an impact on performance in many cases, and that the processor architecture and the gcc compiler's decisions can provoke significant and sometimes surprising variations, in favor of one or the other allocation mode

Integer Affine Transformations of Parametric Z-polytopes and Applications to Loop Nest Optimization

The polyhedral model is a well-known compiler optimization framework for the analysis and transformation of affine loop nests. We present a new method concerning a difficult geometric operation that is raised by this model: the integer affine transformation of parametric Z-polytopes. The result of such a transformation is given by a worst-case exponential union of Z-polytopes. We also propose a polynomial algorithm (for fixed dimension), to count points in arbitrary unions of a fixed number of parametric Z-polytopes. We implemented these algorithms and compared them to other existing algorithms, for a set of applications to loop nest analysis and optimization

The Impact Of The Simulation Using Role-Playing In Developing Resuscitation Care And Leadership Skills For Health Sciences Students : Case Of Future "Midwives"

We would like to sincerely thank the health science studentsfuture "midwives" of the Higher Institute of Nursing and Health Techniques for their participation and availability.Simulation using role-playing is a recurrent didactic strategy among the training activities applied in the field of education. In health sciences training, communication and interaction between students and the various health professionalsinvolved in emergenciesand resuscitation care is one of the strategic pillars of health science education in general and midwifery training in particular. This article reports on the implementation and discusses leadership training in emergency and resuscitation care with simulationusingrole-playing.This approach aimed at increasing awareness of one's own responsibility and that of team member, as it contribute to endurance and development of team management skills in emergency care situations related to resuscitation of pregnant patients in obstetrics units. 19 health science students « future midwives »were involved in the study which was conducted from 20 February to 20 March 2023 at the Higher Institute of Nursing Professions and Health Techniques in Morocco. The results showed that the students were very satisfied with the effectiveness of the role-playing simulation training in improving their professional skills, and feel that they could take on their future function with more ease. Moreover, such a process seems to motivate students to learn and provide them with an individualized approach

Counting integer points in polyhedra and applications to program optimization

Le modèle polyédrique est un formalisme utilisé en optimisation automatique de programmes. Il permet notamment de représenter les itérations et les références à des tableaux, dans des nids de boucles affines, par des points à coordonnées entières de polyèThe polyhedral model is a well-known framework in the field of automatic program optimization. Iterations and array references in affine loop nests are represented by integer points in bounded polyhedra, or (parametric) Z-polytopes. In this thesis, thre

Counting integer points in polyhedra and applications to program optimization

Le modèle polyédrique est un formalisme utilisé en optimisation automatique de programmes. Il permet notamment de représenter les itérations et les références à des tableaux, dans des nids de boucles affines, par des points à coordonnées entières de polyèdres bornés, ou Z-polytopes (paramétrés). Dans cette thèse, trois nouveaux algorithmes de dénombrement ont été développés : des points entiers dans un Z-polytope paramétré, dans une union non disjointe de Z-polytopes paramétrés et dans leurs images par des fonctions affines. Le résultat de ces dénombrements est donné par un ou plusieurs polynômes multivariable à coefficients périodiques. Ces polynômes, connus sous le nom de quasi-polynômes d'Ehrhart, sont définis sur des sous-ensembles de valeurs des paramètres, dits domaines de validité. De nombreuses méthodes d'analyse et d'optimisation de nids de boucles affines font appel à ces algorithmes. Nous les avons en particulier appliqués à la linéarisation de tableaux, dont l'objectif est la compression mémoire et l'amélioration de la localité spatiale. Outre l'optimisation de programmes, les algorithmes proposés ont des applications dans bien d'autres domaines, tels que les mathématiques et l'économie.The polyhedral model is a well-known framework in the field of automatic program optimization. Iterations and array references in affine loop nests are represented by integer points in bounded polyhedra, or (parametric) Z-polytopes. In this thesis, three new counting algorithms have been developed: counting integer points in a parametric Z-polytope, in a union of parametric Z-polytopes and in their images by affine functions. The result of such a counting is given by one or many multivariate polynomials in which the coefficients may be periodic numbers. These polynomials, known as Ehrhart quasipolynomials, are defined on sub-sets of the parameter values called validity domains or chambers. Many affine loop nest analysis and optimization methods require such counting algorithms. We applied them in array linearization which achieves memory compression and improves spatial locality of accessed data. Besides program optimization, the proposed algorithms have many other applications, as in mathematics and economics

Memory optimization by counting points in integer transformations of parametric polytopes

Memory size reduction and memory accesses optimization are crucial issues for embedded systems. In the context of affine programs, these two challenges are classically tackled by array linearization, cache access optimization and memory size computation. Their formalization in the polyhedral model reduce to solving the following problem: count the number of solutions of a Presburger formula. In this paper we propose a novel algorithm that answers this question. We solve the Presburger formula whose solution is a union of parametric Z-polytopes and we propose an algorithm to count points in such a union of parametric Z-polytopes. These algorithms were implemented and we compare them to other existing methods

Méthodes de dénombrement de points entiers de polyèdres et applications à l'optimisation de programmes

Le modèle polyédrique est un formalisme utilisé en optimisation automatique de programmes. Il permet notamment de représenter les itérations et les références à des tableaux, dans des nids de boucles affines, par des points à coordonnées entières de polyèdres bornés, ou Z-polytopes (paramétrés). Dans cette thèse, trois nouveaux algorithmes de dénombrement ont été développés : des points entiers dans un Z-polytope paramétré, dans une union non disjointe de Z-polytopes paramétrés et dans leurs images par des fonctions affines. Le résultat de ces dénombrements est donné par un ou plusieurs polynômes multivariable à coefficients périodiques. Ces polynômes, connus sous le nom de quasi-polynômes d’Ehrhart, sont définis sur des sous-ensembles de valeurs des paramètres, dits domaines de validité. De nombreuses méthodes d’analyse et d’optimisation de nids de boucles affines font appel à ces algorithmes. Nous les avons en particulier appliqués à la linéarisation de tableaux, dont l’objectif est la compression mémoire et l’amélioration de la localité spatiale. Outre l’optimisation de programmes, les algorithmes proposés ont des applications dans bien d’autres domaines, tels que les mathématiques et l’économie.The polyhedral model is a well-known framework in the field of automatic program optimization. Iterations and array references in affine loop nests are represented by integer points in bounded polyhedra, or (parametric) Z-polytopes. In this thesis, three new counting algorithms have been developed: counting integer points in a parametric Z-polytope, in a union of parametric Z-polytopes and in their images by affine functions. The result of such a counting is given by one or many multivariate polynomials in which the coefficients may be periodic numbers. These polynomials, known as Ehrhart quasipolynomials, are defined on sub-sets of the parameter values called validity domains or chambers. Many affine loop nest analysis and optimization methods require such counting algorithms. We applied them in array linearization which achieves memory compression and improves spatial locality of accessed data. Besides program optimization, the proposed algorithms have many other applications, as in mathematics and economics

Détermination des caractéristiques probabilistes du comportement dynamique d'un véhicule.(choix rationnel des paramètres de la suspension)

Colloque avec actes et comité de lecture. Internationale.International audienceLes perturbations provenant de la part de la chaussée dont le caractère est aléatoires (non contrôlables) sont à l'origine de beaucoup d'anomalies mécaniques, le malaise, ainsi que la fatigue des usagers. La mécanique classique, basée sur la notion de déterminisme, n'est plus suffisante pour prendre en considération toutes les influences non contrôlables et d'expliquer le phénomène physique. Dans ce présent travail, afin d'analyser le comportement dynamique du véhicule nous utilisons la théorie spectrale permettant d'obtenir les caractéristiques probabilistes de sortie pour des caractéristiques connues de l'entrée. Quant à la difficulté rencontrée (calcul des éléments de la matrice des fonctions de transfert sur calculateur) nous la contournons par une procédure numérique simple que nous présentons

Détermination des caractéristiques probabilistes du comportement dynamique d'un véhicule.(choix rationnel des paramètres de la suspension)

Les perturbations provenant de la part de la chaussée dont le caractère est aléatoires (non contrôlables) sont à l'origine de beaucoup d'anomalies mécaniques, le malaise, ainsi que la fatigue des usagers. La mécanique classique, basée sur la notion de déterminisme, n'est plus suffisante pour prendre en considération toutes les influences non contrôlables et d'expliquer le phénomène physique. Dans ce présent travail, afin d'analyser le comportement dynamique du véhicule nous utilisons la théorie spectrale permettant d'obtenir les caractéristiques probabilistes de sortie pour des caractéristiques connues de l'entrée. Quant à la difficulté rencontrée (calcul des éléments de la matrice des fonctions de transfert sur calculateur) nous la contournons par une procédure numérique simple que nous présentons