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.Key-words: sparse matrices, parallel multifrontal method, dynamic scheduling, memory.This text is also available as a research report of the Laboratoire de l'Informatique du Parallélisme http://www.ens-lyon.fr/LIP and as a technical report from ENSEEIHT-IRIT. Stratégies d'ordonnancement hybrides pour la résolution parallèle de systèmes linéairesRésumé : Nous proposons des stratégies d'ordonnancement bi-critères, qui s'inté-ressentà la foisà la performance età 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 aété implanté, qui prend en compte ces nouvelles contraintes. De plus, la performance aété améliorée parce que notre nouvelle approche permet a l'ordonnanceur de prendre des décisions meilleures, quiétaient 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 rapportà l'approche initiale.

To face the advent of multicore processors and the ever increasing complexity of hardware architectures, programming models based on DAG parallelism regained popularity in the high performance, scientific computing community. Modern runtime systems offer a programming interface that complies with this paradigm and powerful engines for scheduling the tasks into which the application is decomposed. These tools have already proved their effectiveness on a number of dense linear algebra applications. This paper evaluates the usability and effectiveness of runtime systems based on the Sequential Task Flow model for complex applications, namely, sparse matrix multifrontal factorizations which feature extremely irregular workloads, with tasks of different granularities and characteristics and with a variable memory consumption. Most importantly, it shows how this parallel programming model eases the development of complex features that benefit the performance of sparse, direct solvers as well as their memory consumption. We illustrate our discussion with the multifrontal QR factorization running on top of the StarPU runtime system.

To face the advent of multicore processors and the ever increasing complexity of hardware architectures, programming models based on DAG parallelism regained popularity in the high performance, scientific computing community. Modern runtime systems offer a programming interface that complies with this paradigm and powerful engines for scheduling the tasks into which the application is decomposed. These tools have already proved their effectiveness on a number of dense linear algebra applications. This paper evaluates the usability of runtime systems for complex applications, namely, sparse matrix multifrontal factorizations which constitute extremely irregular workloads, with tasks of different granularities and characteristics and with a variable memory consumption. Experimental results on real-life matrices show that it is possible to achieve the same efficiency as with an ad hoc scheduler which relies on the knowledge of the algorithm. A detailed analysis shows the performance behavior of the resulting code and possible ways of improving the effectiveness of runtime systems.

Abstract. We 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.

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We are interested in the memory usage of multifrontal methods. Starting from the algorithms introduced by Liu, we propose new schedules to allocate and process tasks that improve memory usage. This generalizes two existing factorization and memory-allocation schedules by allowing a more flexible task allocation together with a specific tree traversal. We present optimal algorithms for this new class of schedules, and demonstrate experimentally their benefit for some real-world matrices from sparse matrix collections where either the active memory or the total memory is minimized.

Enabling HPC applications to perform efficiently when invoking multiple parallel libraries simultaneously is a great challenge. Even if a single runtime system is used underneath, scheduling tasks or threads coming from different libraries over the same set of hardware resources introduces many issues, such as resource oversubscription, undesirable cache flushes or memory bus contention.This paper presents an extension of StarPU, a runtime system specifically designed for heterogeneous architectures, that allows multiple parallel codes to run concurrently with minimal interference. Such parallel codes run within scheduling contexts that provide confined execution environments which can be used to partition computing resources. Scheduling contexts can be dynamically resized to optimize the allocation of computing resources among concurrently running libraries. We introduce a hypervisor that automatically expands or shrinks contexts using feedback from the runtime system (e.g. resource utilization). We demonstrate the relevance of our approach using benchmarks invoking multiple high performance linear algebra kernels simultaneously on top of heterogeneous multicore machines. We show that our mechanism can dramatically improve the overall application run time (-34%), most notably by reducing the average cache miss ratio (-50%).

International audienceAs the computational power of high performance computing (HPC) systems continues to increase by using a huge number of cores or specialized processing units, HPC applications are increasingly prone to faults. In this paper, we present a new class of numerical fault tolerance algorithms to cope with node crashes in parallel distributed environments. This new resilient scheme is designed at application level and does not require extra resources, i.e., computational unit or computing time, when no fault occurs. In the framework of iterative methods for the solution of sparse linear systems, we present numerical algorithms to extract relevant information from available data after a fault, assuming a separate mechanism ensures the fault detection. After data extraction, a well chosen part of missing data is regenerated through interpolation strategies to constitute meaningful inputs to restart the iterative scheme. We have developed these methods, referred to as Interpolation-Restart techniques, for Krylov subspace linear solvers. After a fault, lost entries of the current iterate computed by the solver are interpolated to define a new initial guess to restart the Krylov method. A well suited initial guess is computed by using the entries of the faulty iterate available on surviving nodes. We present two interpolation policies that preserve key numerical properties of well-known linear solvers, namely the monotonic decrease of the A-norm of the error of the conjugate gradient or the residual norm decrease of GMRES. The qualitative numerical behavior of the resulting scheme have been validated with sequential simulations, when the number of faults and the amount of data losses are varied. Finally, the computational costs associated with the recovery mechanism have been evaluated through parallel experiments