Sur la conception de solveurs linéaires hybrides pour les architectures parallèles modernes

In the context of this thesis, our focus is on numerical linear algebra, more precisely on solution of large sparse systems of linear equations. We focus on designing efficient parallel implementations of MaPHyS, an hybrid linear solver based on domain decomposition techniques. First we investigate the MPI+threads approach. In MaPHyS, the first level of parallelism arises from the independent treatment of the various subdomains. The second level is exploited thanks to the use of multi-threaded dense and sparse linear algebra kernels involved at the subdomain level. Such an hybrid implementation of an hybrid linear solver suitably matches the hierarchical structure of modern supercomputers and enables a trade-off between the numerical and parallel performances of the solver. We demonstrate the flexibility of our parallel implementation on a set of test examples. Secondly, we follow a more disruptive approach where the algorithms are described as sets of tasks with data inter-dependencies that leads to a directed acyclic graph (DAG) representation. The tasks are handled by a runtime system. We illustrate how a first task-based parallel implementation can be obtained by composing task-based parallel libraries within MPI processes throught a preliminary prototype implementation of our hybrid solver. We then show how a task-based approach fully abstracting the hardware architecture can successfully exploit a wide range of modern hardware architectures. We implemented a full task-based Conjugate Gradient algorithm and showed that the proposed approach leads to very high performance on multi-GPU, multicore and heterogeneous architectures.Dans le contexte de cette thèse, nous nous focalisons sur des algorithmes pour l’algèbre linéaire numérique, plus précisément sur la résolution de grands systèmes linéaires creux. Nous mettons au point des méthodes de parallélisation pour le solveur linéaire hybride MaPHyS. Premièrement nous considerons l'aproche MPI+threads. Dans MaPHyS, le premier niveau de parallélisme consiste au traitement indépendant des sous-domaines. Le second niveau est exploité grâce à l’utilisation de noyaux multithreadés denses et creux au sein des sous-domaines. Une telle implémentation correspond bien à la structure hiérarchique des supercalculateurs modernes et permet un compromis entre les performances numériques et parallèles du solveur. Nous démontrons la flexibilité de notre implémentation parallèle sur un ensemble de cas tests. Deuxièmement nous considérons un approche plus innovante, où les algorithmes sont décrits comme des ensembles de tâches avec des inter-dépendances, i.e., un graphe de tâches orienté sans cycle (DAG). Nous illustrons d’abord comment une première parallélisation à base de tâches peut être obtenue en composant des librairies à base de tâches au sein des processus MPI illustrer par un prototype d’implémentation préliminaire de notre solveur hybride. Nous montrons ensuite comment une approche à base de tâches abstrayant entièrement le matériel peut exploiter avec succès une large gamme d’architectures matérielles. À cet effet, nous avons implanté une version à base de tâches de l’algorithme du Gradient Conjugué et nous montrons que l’approche proposée permet d’atteindre une très haute performance sur des architectures multi-GPU, multicoeur ainsi qu’hétérogène

Task-based hybrid linear solver for distributed memory heterogeneous architectures

Heterogeneity is emerging as one of the most challenging characteristics of today’s parallel environments. However, not many fully-featured advanced numerical, scientific libraries have been ported on such architectures. In this paper, we propose to extend a sparse hybrid solver for handling distributed memory heterogeneous platforms. As in the original solver, we perform a domain decomposition and associate one subdomain with one MPI process. However, while each subdomain was processed sequentially (binded onto a single CPU core) in the original solver, the new solver instead relies on task-based local solvers, delegating tasks to available computing units. We show that this “MPI+task” design conveniently allows for exploiting distributed memory heterogeneous machines. Indeed, a subdomain can now be processed on multiple CPU cores (such as a whole multicore processor or a subset of the available cores) possibly enhanced with GPUs. We illustrate our discussion with the MaPHyS sparse hybrid solver relying on the PaStiX and Chameleon dense and sparse direct libraries, respectively. Interestingly, this two-level MPI+task design furthermore provides extra flexibility for controlling the number of subdomains, enhancing the numerical stability of the considered hybrid method. While the rise of heterogeneous computing has been strongly carried out by the theoretical community, this study aims at showing that it is now also possible to build complex software layers on top of runtime systems to exploit heterogeneous architectures

Hierarchical hybrid sparse linear solver for multicore platforms

The solution of large sparse linear systems is a critical operationfor many numerical simulations. To cope with the hierarchical designof modern supercomputers, hybrid solvers based on Domain DecompositionMethods (DDM) have been been proposed. Among them, approachesconsisting of solving the problem on the interior of the domains witha sparse direct method and the problem on their interface with apreconditioned iterative method applied to the related SchurComplement have shown an attractive potential as they can combine therobustness of direct methods and the low memory footprint of iterativemethods. In this report, we consider an additive Schwarz preconditionerfor the Schur Complement, which represents a scalable candidate butwhose numerical robustness may decrease when the number of domainsbecomes too large. We thus propose a two-level MPI/thread parallelapproach to control the number of domains and hence the numericalbehaviour. We illustrate our discussion with large-scale matricesarising from real-life applications and processed on both a moderncluster and a supercomputer. We show that the resulting method canprocess matrices such as tdr455k for which we previously either ranout of memory on few nodes or failed to converge on a larger number ofnodes. Matrices such as Nachos_4M that could not be correctly processedin the past can now be efficiently processed up to a very large numberof CPU cores (24,576 cores). The corresponding code has beenincorporated into the MaPHyS package

Task-based Conjugate-Gradient for multi-GPUs platforms

Whereas most today parallel High Performance Computing (HPC) software is written as highly tuned code taking care of low-level details, the advent of the manycore area forces the community to consider modular programming paradigms and delegate part of the work to a third party software. That latter approach has been shown to be very productive and efficient with regular algorithms, such as dense linear algebra solvers. In this paper we show that such a model can be efficiently applied to a much more irregular and less compute intensive algorithm. We illustrate our discussion with the standard unpreconditioned Conjugate Gradient (CG) that we carefully express as a task-based algorithm. We use the StarPU runtime system to assess the efficiency of the approach on a computational platform consisting of three NVIDIA Fermi GPUs. We show that almost optimum speed up (up to 2.89) may be reached (relatively to a mono-GPU execution) when processing large matrices and that the performance is portable when changing the low-level memory transfer mechanism.andis que la plupart des logiciels de calcul haute performance (HPC) actuels sont des codes extrêmement optimisés en prenant en compte les détails de bas-niveau, l'avènement de l'ère manycore incite la communauté à considèrer des paradigmes de programmation mod- ulaires et ainsi déléguer une partie du travail à des librairies tierces. Cette dernière approche s'est avérée très productive et efficace dans le cas d'algorithmes réguliers, tels que ceux issus de l'algèbre linéaire dense. Dans ce papier, nous démontrons qu'un tel modèle peut être effi- cacement appliqué à un problème beaucoup plus irrégulier et moins intensif en calcul. Nous illustrons notre discussion avec l'algorithme standard du Gradient Conjugué (CG) non précon- ditionné que nous exprimons sous la forme d'un algorithme en graphe de tâches. Nous utilisons le moteur d'exécution StarPU pour évaluer l'efficacité de notre approche sur une plate-forme de calcul composée de trois accélérateurs graphiques (GPU) NVIDIA Fermi. Nous démontrons qu'une accroissement de performance (jusqu'à un facteur 2, 89) quasi optimal (relativement au cas mono-GPU) peut être atteinte lorsque sont traitées des matrices creuses de grande taille. Nous montrons de surcroît que la performance est portable quand les mécanismes de transfert mémoire bas-niveau sont changés

On the design of sparse hybrid linear solvers for modern parallel architectures

Dans le contexte de cette thèse, nous nous focalisons sur des algorithmes pour l’algèbre linéaire numérique, plus précisément sur la résolution de grands systèmes linéaires creux. Nous mettons au point des méthodes de parallélisation pour le solveur linéaire hybride MaPHyS. Premièrement nous considerons l'aproche MPI+threads. Dans MaPHyS, le premier niveau de parallélisme consiste au traitement indépendant des sous-domaines. Le second niveau est exploité grâce à l’utilisation de noyaux multithreadés denses et creux au sein des sous-domaines. Une telle implémentation correspond bien à la structure hiérarchique des supercalculateurs modernes et permet un compromis entre les performances numériques et parallèles du solveur. Nous démontrons la flexibilité de notre implémentation parallèle sur un ensemble de cas tests. Deuxièmement nous considérons un approche plus innovante, où les algorithmes sont décrits comme des ensembles de tâches avec des inter-dépendances, i.e., un graphe de tâches orienté sans cycle (DAG). Nous illustrons d’abord comment une première parallélisation à base de tâches peut être obtenue en composant des librairies à base de tâches au sein des processus MPI illustrer par un prototype d’implémentation préliminaire de notre solveur hybride. Nous montrons ensuite comment une approche à base de tâches abstrayant entièrement le matériel peut exploiter avec succès une large gamme d’architectures matérielles. À cet effet, nous avons implanté une version à base de tâches de l’algorithme du Gradient Conjugué et nous montrons que l’approche proposée permet d’atteindre une très haute performance sur des architectures multi-GPU, multicoeur ainsi qu’hétérogène.In the context of this thesis, our focus is on numerical linear algebra, more precisely on solution of large sparse systems of linear equations. We focus on designing efficient parallel implementations of MaPHyS, an hybrid linear solver based on domain decomposition techniques. First we investigate the MPI+threads approach. In MaPHyS, the first level of parallelism arises from the independent treatment of the various subdomains. The second level is exploited thanks to the use of multi-threaded dense and sparse linear algebra kernels involved at the subdomain level. Such an hybrid implementation of an hybrid linear solver suitably matches the hierarchical structure of modern supercomputers and enables a trade-off between the numerical and parallel performances of the solver. We demonstrate the flexibility of our parallel implementation on a set of test examples. Secondly, we follow a more disruptive approach where the algorithms are described as sets of tasks with data inter-dependencies that leads to a directed acyclic graph (DAG) representation. The tasks are handled by a runtime system. We illustrate how a first task-based parallel implementation can be obtained by composing task-based parallel libraries within MPI processes throught a preliminary prototype implementation of our hybrid solver. We then show how a task-based approach fully abstracting the hardware architecture can successfully exploit a wide range of modern hardware architectures. We implemented a full task-based Conjugate Gradient algorithm and showed that the proposed approach leads to very high performance on multi-GPU, multicore and heterogeneous architectures

Task-based sparse hybrid linear solver for distributed memory heterogeneous architectures

International audienc

Combining Software Pipelining with Numerical Pipelining in the Conjugate Gradient Algorithm

International audienc

Task-based Conjugate Gradient: from multi-GPU towards heterogeneous architectures

International audienceWhereas most parallel High Performance Computing (HPC) numerical libaries have been written as highly tuned and mostly monolithic codes, the increased complexity of modern architectures led the computational science and engineering community to consider more modular programming paradigms such as task-based paradigms to design new generation of parallel simulation code; this enables to delegate part of the work to a third party software such as a runtime system. That latter approach has been shown to be very productive and efficient with compute-intensive algorithms, such as dense linear algebra and sparse direct solvers. In this study, we consider a much more irregular, and synchronizing algorithm, namely the Conjugate Gradient (CG) algorithm. We propose a task-based formulation of the algorithm together with a very fine instrumentation of the runtime system. We show that almost optimum speed up may be reached on a multi-GPU platform (relatively to the mono-GPU case) and, as a very preliminary but promising result, that the approach can be effectively used to handle heterogenous architectures composed of a multicore chip and multiple GPUs. We expect that these results will pave the way for investigating the design of new advanced, irregular numerical algorithms on top of runtime systems

Pipelining the CG Solver Over a Runtime System

International audienceWhereas most today parallel High Performance Computing (HPC) software is written as highly tuned code taking care of low-level details, the advent of the manycore area forces the community to consider modular programming paradigms and delegate part of the work to a third party software. That latter approach has been shown to be very productive and efficient with regular algorithms, such as dense linear algebra solvers. In this paper we show that such a model can be efficiently applied to a much more irregular and less compute intensive algorithm. We illustrate our discussion with the standard unpreconditioned Conjugate Gradient (CG) that we carefully express as a task-based algorithm. We use the StarPU runtime system to assess the efficiency of the approach on a computational platform consisting of three NVIDIA Fermi GPUs. We show that almost optimum speed up (up to 2.89) may be reached (relatively to a mono-GPU execution) when processing large matrices and that the performance is portable when changing the low-level memory transfer mechanism

On a hierarchical parallel algebraic domain decomposition linear solver

International audienc