Fast and Accurate Simulation of Multithreaded Sparse Linear Algebra Solvers

International audienceThe ever growing complexity and scale of parallel architectures imposes to rewrite classical monolithic HPC scientific applications and libraries as their portability and performance optimization only comes at a prohibitive cost. There is thus a recent and general trend in using instead a modular approach where numerical algorithms are written at a high level independently of the hardware architecture as Directed Acyclic Graphs (DAG) of tasks. A task-based runtime system then dynamically schedules the resulting DAG on the different computing resources, automatically taking care of data movement and taking into account the possible speed heterogeneity and variability. Evaluating the performance of such complex and dynamic systems is extremely challenging especially for irregular codes. In this article, we explain how we crafted a faithful simulation, both in terms of performance and memory usage, of the behavior of qr_mumps, a fully-featured sparse linear algebra library, on multi-core architectures. In our approach, the target high-end machines are calibrated only once to derive sound performance models. These models can then be used at will to quickly predict and study in a reproducible way the performance of such irregular and resource-demanding applications using solely a commodity laptop

A Makespan Lower Bound for the Scheduling of the Tiled Cholesky Factorization based on ALAP Schedule

International audienceDue to the advent of multicore architectures and massive parallelism, the tiled Cholesky factorization algorithm has recently received plenty of attention and is often referenced by practitioners as a case study. It is also implemented in mainstream dense linear algebra libraries and is used as a testbed for runtime systems. However, we note that theoretical study of the parallelism of this algorithm is currently lacking. In this paper, we present new theoretical results about the tiled Cholesky factorization in the context of a parallel homogeneous model without communication costs. Based on the relative costs of involved kernels, we prove that only two different situations must be considered, typically corresponding to CPUs and GPUs. By a careful analysis on the number of tasks of each type that run simultaneously in the ALAP (As Late As Possible) schedule without resource limitation, we are able to determine precisely the number of busy processors at any time (as degree 2 polynomials). We then use this information to find a closed form formula for the minimum time to schedule a tiled Cholesky factorization of size n on P processors. We show that this bound outperforms classical bounds from the literature. We also prove that ALAP(P), an ALAP-based schedule where the number of resources is limited to P , has a makespan extremely close to the lower bound, thus proving both the effectiveness of ALAP(P) schedule and of the lower bound on the makespan

A Visual Performance Analysis Framework for Task-based Parallel Applications running on Hybrid Clusters

International audienceProgramming paradigms in High-Performance Computing have been shifting towards task-based models which are capable of adapting readily to heterogeneous and scalable supercomputers. The performance of task-based application heavily depends on the runtime scheduling heuristics and on its ability to exploit computing and communication resources. Unfortunately, the traditional performance analysis strategies are unfit to fully understand task-based runtime systems and applications: they expect a regular behavior with communication and computation phases, while task-based applications demonstrate no clear phases. Moreover, the finer granularity of task-based applications typically induces a stochastic behavior that leads to irregular structures that are difficult to analyze. Furthermore, the combination of application structure, scheduler, and hardware information is generally essential to understand performance issues. This paper presents a flexible framework that enables one to combine several sources of information and to create custom visualization panels allowing to understand and pinpoint performance problems incurred by bad scheduling decisions in task-based applications. Three case-studies using StarPU-MPI, a task-based multi-node runtime system, are detailed to show how our framework can be used to study the performance of the well-known Cholesky factorization. Performance improvements include a better task partitioning among the multi-(GPU,core) to get closer to theoretical lower bounds, improved MPI pipelining in multi-(node,core,GPU) to reduce the slow start, and changes in the runtime system to increase MPI bandwidth, with gains of up to 13% in the total makespan

2D Static Resource Allocation for Compressed Linear Algebra and Communication Constraints

International audienceThis paper adresses static resource allocation problems for irregular distributed parallel applications. More precisely, we focus on two classical tiled linear algebra kernels: the Matrix Multiplication (MM) and the LU decomposition (LU) algorithms on large linear systems. In the context of parallel distributed platforms, data exchanges can dramatically degrade the performance of linear algebra kernels and in this context, compression techniques such as Block Low Rank (BLR) compression techniques are good candidates both for limiting data storage on each processor and data exchanges between processors. On the other hand, the use of BLR representation makes the static allocation problem of tiles to processors more complex. Indeed, the load associated to each tile depends on its compression factor, which induces an heterogeneous load balancing problem. In turn, solving this load balancing problem optimally might lead to complex allocation schemes, where the tiles allocated to a given processor are scattered all over the matrix. This in turn induces communication costs, since matrix multiplication and LU decompositions heavily rely on broadcasting operations along rows and columns of processors, so that the communication volume is minimized when the maximal number of different processors on any row and column is minimized. In the fully homogeneous case, 2D Block Cyclic (BC) allocation solves both load balancing and communication minimization issues simultaneously , but it might lead to bad load balancing in the heterogeneous case. Our goal in this paper is to propose data allocation schemes dedicated to BLR format and to prove that it is possible to obtain good performance on makespan when simultaneously balancing the load and minimizing the maximal number of different processor in any row or column

Task-based Runtime Optimizations Towards High Performance Computing Applications

The last decades have witnessed a rapid improvement of computational capabilities in high-performance computing (HPC) platforms thanks to hardware technology scaling. HPC architectures benefit from mainstream advances on the hardware with many-core systems, deep hierarchical memory subsystem, non-uniform memory access, and an ever-increasing gap between computational power and memory bandwidth. This has necessitated continuous adaptations across the software stack to maintain high hardware utilization. In this HPC landscape of potentially million-way parallelism, task-based programming models associated with dynamic runtime systems are becoming more popular, which fosters developers’ productivity at extreme scale by abstracting the underlying hardware complexity. In this context, this dissertation highlights how a software bundle powered by a task-based programming model can address the heterogeneous workloads engendered by HPC applications., i.e., data redistribution, geospatial modeling and 3D unstructured mesh deformation here. Data redistribution aims to reshuffle data to optimize some objective for an algorithm, whose objective can be multi-dimensional, such as improving computational load balance or decreasing communication volume or cost, with the ultimate goal of increasing the efficiency and therefore reducing the time-to-solution for the algorithm. Geostatistical modeling, one of the prime motivating applications for exascale computing, is a technique for predicting desired quantities from geographically distributed data, based on statistical models and optimization of parameters. Meshing the deformable contour of moving 3D bodies is an expensive operation that can cause huge computational challenges in fluid-structure interaction (FSI) applications. Therefore, in this dissertation, Redistribute-PaRSEC, ExaGeoStat-PaRSEC and HiCMA-PaRSEC are proposed to efficiently tackle these HPC applications respectively at extreme scale, and they are evaluated on multiple HPC clusters, including AMD-based, Intel-based, Arm-based CPU systems and IBM-based multi-GPU system. This multidisciplinary work emphasizes the need for runtime systems to go beyond their primary responsibility of task scheduling on massively parallel hardware system for servicing the next-generation scientific applications

Gestión de recursos energéticamente eficiente para aplicaciones paralelas basadas en tareas en entornos multi-aplicación

Tesis de la Universidad Complutense de Madrid, Facultad de Informática, leída el 28/01/2021The end of Dennard scaling, as well as the arrival of the post-Moore era, has meant a big change in the way performance and energy efficiency are achieved by modern processors. From a constant increase of the clock frequency as the main method to increase performance at the beginning of the 2000s, the increase in the number of cores inside processors running at relatively conservative frequencies has stabilised as the current trend to increase both performance and energy efficiency. The increase of the heterogeneity in the systems, both inside the processors comprising different types of cores (e.g., big LITTLE architectures) or adding specific compute units (like multimedia extensions), as well as in the platform by the addition of other specific compute units (like GPUs), offering different performance and energy-efficiency trade-offs. Together with the increase in the number of cores, the processor evolution has been accompanied by the addition of different techologies that allow processors to adapt dynamically to the changes in the environment and running aplications. Among others, techiniques like dynamic voltage and frequiency scaling, power capping or cache partitioning are widely used nowadays to increase the performance and/or energy-efficiency...El fin del escalado de Dennard, así como la llegada de la era post-Moore ha supuesto una gran revolución en la forma de obtener el rendimiento y eficiencia energética en los procesadores modernos. Desde un incremento constante en la frecuencia relativamente moderadas se ha impuesto como la tendencia actual para incrementar tanto el rendimiento como la eficiencia energética. El aumento del número de núcleos dentro del procesado ha venido acompañado en los últimos años por el aumento de la heterogeneidad en la plataforma, tanto dentro del procesador incorporando distintos tipos de núcleos en el mismo procesador (e.g., la arquitectura big.LITTLE) como añadiendo unidades de cómputo específicas (e.g., extensiones multimedia), como la incorporación de otros elementos de computo específicos, ofreciendo diferentes grados de rendimiento y eficiencia energética. La evolución de los procesadores no solo ha venido dictada por el aumento del número de núcleos, sino que ha venido acompañada por la incorporación de diferentes técnicas permitiendo la adaptación de las arquitecturas de forma dinámica al entorno así como a las aplicaciones en ejecución. Entre otras, técnicas como el escalado de frecuencia, la limitación de consumo o el particionado de la memoria caché son ampliamente utilizadas en la actualidad como métodos para incrementar el consumo y/o la eficiencia energética...Fac. de InformáticaTRUEunpu

Context adaptivity for selected computational kernels with applications in optoelectronics and in phylogenetics

Computational Kernels sind der kritische Teil rechenintensiver Software, wofür der größte Rechenaufwand anfällt; daher müssen deren Design und Implementierung sorgfältig vorgenommen werden. Zwei wissenschaftliche Anwendungsprobleme aus der Optoelektronik und aus der Phylogenetik, sowie dazugehörige Computational Kernels motivieren diese Arbeit. Im ersten Anwendungsproblem werden Komponenten zur Berechnung komplex-symmetrischer Eigenwertprobleme diskutiert, welche in der Simulation von Wellenleitern in der Optoelektronik auftreten. LAPACK und ScaLAPACK beinhalten sehr leistungsfähige Referenzimplementierungen für bestimmte Problemstellungen der linearen Algebra. In Bezug auf Eigenwertprobleme werden ausschließlich reell-symmetrische und komplex-hermitesche Varianten angeboten, daher sind effiziente Codes für komplex-symmetrische (nicht-hermitesche) Eigenwertprobleme sehr wünschenswert. Das zweite Anwendungsproblem behandelt einen parallelen, wissenschaftlichen Workflow zur Rekonstruktion von Phylogenien, welcher entworfen, umgesetzt und evaluiert wird. Die Rekonstruktion von phylogenetischen Bäumen ist ein NP-hartes Problem, welches äußerst viel Rechenkapazität benötigt, wodurch ein paralleler Ansatz erforderlich ist. Die grundlegende Idee dieser Arbeit ist die Untersuchung der Wechselbeziehung zwischen dem Kontext der behandelten Kernels und deren Effizienz. Ein Kontext eines Computational Kernels beinhaltet Modellaspekte (z.B. Struktur der Eingabedaten), Softwareaspekte (z.B. rechenintensive Bibliotheken), Hardwareaspekte (z.B. verfügbarer Hauptspeicher und unterstützte darstellbare Genauigkeit), sowie weitere Anforderungen bzw. Einschränkungen. Einschränkungen sind hinsichtlich Laufzeit, Speicherverbrauch, gelieferte Genauigkeit usw., möglich. Das Konzept der Kontextadaptivität wird für ausgewählte Anwendungsprobleme in Computational Science gezeigt. Die vorgestellte Methode ist ein Meta-Algorithmus, der Aspekte des Kontexts verwendet, um optimale Leistung hinsichtlich der angewandten Metrik zu erzielen. Es ist wichtig, den Kontext einzubeziehen, weil Anforderungen gegeneinander ausgetauscht werden könnten, resultierend in einer höheren Leistung. Zum Beispiel kann im Falle einer niedrigen benötigten Genauigkeit ein schnellerer Algorithmus einer bewährten, aber langsameren, Methode vorgezogen werden. Speziell für komplex-symmetrische Eigenwertprobleme zugeschnittene Codes zielen darauf ab, Genauigkeit gegen Geschwindigkeit einzutauschen. Die Innovation wird durch neue algorithmische Ansätze belegt, welche die algebraische Struktur ausnutzen. Bezüglich der Berechnung von phylogenetischen Bäumen wird die Abbildung eines Workflows auf ein Campusgrid-System gezeigt. Die Innovation besteht in der anpassungsfähigen Implementierung des Workflows, der nebenläufige Instanzen von Computational Kernels in einem verteilten System darstellt. Die Adaptivität bezeichnet hier die Fähigkeit des Workflows, die Rechenlast hinsichtlich verfügbarer Rechner, Zeit und Qualität der phylogenetischen Bäume anzupassen. Kontextadaptivität wird durch die Implementierung und Evaluierung von wissenschaftlichen Problemstellungen aus der Optoelektronik und aus der Phylogenetik gezeigt. Für das Fachgebiet der Optoelektronik zielt eine Familie von Algorithmen auf die Lösung von verallgemeinerten komplex-symmetrischen Eigenwertproblemen ab. Unser alternativer Ansatz nutzt die symmetrische Struktur aus und spielt günstigere Laufzeit gegen eine geringere Genauigkeit aus. Dieser Ansatz ist somit schneller, jedoch (meist) ungenauer als der konventionelle Lösungsweg. Zusätzlich zum sequentiellen Löser wird eine parallele Variante diskutiert und teilweise auf einem Cluster mit bis zu 1024 CPU-Cores evaluiert. Die erzielten Laufzeiten beweisen die Überlegenheit unseres Ansatzes -- allerdings sind weitere Untersuchungen zur Erhöhung der Genauigkeit notwendig. Für das Fachgebiet der Phylogenetik zeigen wir, dass die phylogenetische Baum-Rekonstruktion mittels eines Condor-basierten Campusgrids effizient parallelisiert werden kann. Dieser parallele wissenschaftliche Workflow weist einen geringen parallelen Overhead auf, resultierend in exzellenter Effizienz.Computational kernels are the crucial part of computationally intensive software, where most of the computing time is spent; hence, their design and implementation have to be accomplished carefully. Two scientific application problems from optoelectronics and from phylogenetics and corresponding computational kernels are motivating this thesis. In the first application problem, components for the computational solution of complex symmetric EVPs are discussed, arising in the simulation of waveguides in optoelectronics. LAPACK and ScaLAPACK contain highly effective reference implementations for certain numerical problems in linear algebra. With respect to EVPs, only real symmetric and complex Hermitian codes are available, therefore efficient codes for complex symmetric (non-Hermitian) EVPs are highly desirable. In the second application problem, a parallel scientific workflow for computing phylogenies is designed, implemented, and evaluated. The reconstruction of phylogenetic trees is an NP-hard problem that demands huge scale computing capabilities, and therefore a parallel approach is necessary. One idea underlying this thesis is to investigate the interaction between the context of the kernels considered and their efficiency. The context of a computational kernel comprises model aspects (for instance, structure of input data), software aspects (for instance, computational libraries), hardware aspects (for instance, available RAM and supported precision), and certain requirements or constraints. Constraints may exist with respect to runtime, memory usage, accuracy required, etc.. The concept of context adaptivity is demonstrated to selected computational problems in computational science. The method proposed here is a meta-algorithm that utilizes aspects of the context to result in an optimal performance concerning the applied metric. It is important to consider the context, because requirements may be traded for each other, resulting in a higher performance. For instance, in case of a low required accuracy, a faster algorithmic approach may be favored over an established but slower method. With respect to EVPs, prototypical codes that are especially targeted at complex symmetric EVPs aim at trading accuracy for speed. The innovation is evidenced by the implementation of new algorithmic approaches exploiting structure. Concerning the computation of phylogenetic trees, the mapping of a scientific workflow onto a campus grid system is demonstrated. The adaptive implementation of the workflow features concurrent instances of a computational kernel on a distributed system. Here, adaptivity refers to the ability of the workflow to vary computational load in terms of available computing resources, available time, and quality of reconstructed phylogenetic trees. Context adaptivity is discussed by means of computational problems from optoelectronics and from phylogenetics. For the field of optoelectronics, a family of implemented algorithms aim at solving generalized complex symmetric EVPs. Our alternative approach exploiting structural symmetry trades runtime for accuracy, hence, it is faster but (usually) features a lower accuracy than the conventional approach. In addition to a complete sequential solver, a parallel variant is discussed and partly evaluated on a cluster utilizing up to 1024 CPU cores. Achieved runtimes evidence the superiority of our approach, however, further investigations on improving accuracy are suggested. For the field of phylogenetics, we show that phylogenetic tree reconstruction can efficiently be parallelized on a campus grid infrastructure. The parallel scientific workflow features a moderate parallel overhead, resulting in an excellent efficiency

Custom optimization algorithms for efficient hardware implementation

The focus is on real-time optimal decision making with application in advanced control systems. These computationally intensive schemes, which involve the repeated solution of (convex) optimization problems within a sampling interval, require more efficient computational methods than currently available for extending their application to highly dynamical systems and setups with resource-constrained embedded computing platforms. A range of techniques are proposed to exploit synergies between digital hardware, numerical analysis and algorithm design. These techniques build on top of parameterisable hardware code generation tools that generate VHDL code describing custom computing architectures for interior-point methods and a range of first-order constrained optimization methods. Since memory limitations are often important in embedded implementations we develop a custom storage scheme for KKT matrices arising in interior-point methods for control, which reduces memory requirements significantly and prevents I/O bandwidth limitations from affecting the performance in our implementations. To take advantage of the trend towards parallel computing architectures and to exploit the special characteristics of our custom architectures we propose several high-level parallel optimal control schemes that can reduce computation time. A novel optimization formulation was devised for reducing the computational effort in solving certain problems independent of the computing platform used. In order to be able to solve optimization problems in fixed-point arithmetic, which is significantly more resource-efficient than floating-point, tailored linear algebra algorithms were developed for solving the linear systems that form the computational bottleneck in many optimization methods. These methods come with guarantees for reliable operation. We also provide finite-precision error analysis for fixed-point implementations of first-order methods that can be used to minimize the use of resources while meeting accuracy specifications. The suggested techniques are demonstrated on several practical examples, including a hardware-in-the-loop setup for optimization-based control of a large airliner.Open Acces