Batched Linear Algebra Problems on GPU Accelerators

The emergence of multicore and heterogeneous architectures requires many linear algebra algorithms to be redesigned to take advantage of the accelerators, such as GPUs. A particularly challenging class of problems, arising in numerous applications, involves the use of linear algebra operations on many small-sized matrices. The size of these matrices is usually the same, up to a few hundred. The number of them can be thousands, even millions. Compared to large matrix problems with more data parallel computation that are well suited on GPUs, the challenges of small matrix problems lie in the low computing intensity, the large sequential operation fractions, and the big PCI-E overhead. These challenges entail redesigning the algorithms instead of merely porting the current LAPACK algorithms. We consider two classes of problems. The first is linear systems with one-sided factorizations (LU, QR, and Cholesky) and their solver, forward and backward substitution. The second is a two-sided Householder bi-diagonalization. They are challenging to develop and are highly demanded in applications. Our main efforts focus on the same-sized problems. Variable-sized problems are also considered, though to a lesser extent. Our contributions can be summarized as follows. First, we formulated a batched linear algebra framework to solve many data-parallel, small-sized problems/tasks. Second, we redesigned a set of fundamental linear algebra algorithms for high- performance, batched execution on GPU accelerators. Third, we designed batched BLAS (Basic Linear Algebra Subprograms) and proposed innovative optimization techniques for high-performance computation. Fourth, we illustrated the batched methodology on real-world applications as in the case of scaling a CFD application up to 4096 nodes on the Titan supercomputer at Oak Ridge National Laboratory (ORNL). Finally, we demonstrated the power, energy and time efficiency of using accelerators as compared to CPUs. Our solutions achieved large speedups and high energy efficiency compared to related routines in CUBLAS on NVIDIA GPUs and MKL on Intel Sandy-Bridge multicore CPUs. The modern accelerators are all Single-Instruction Multiple-Thread (SIMT) architectures. Our solutions and methods are based on NVIDIA GPUs and can be extended to other accelerators, such as the Intel Xeon Phi and AMD GPUs based on OpenCL

Resilience for Asynchronous Iterative Methods for Sparse Linear Systems

Large scale simulations are used in a variety of application areas in science and engineering to help forward the progress of innovation. Many spend the vast majority of their computational time attempting to solve large systems of linear equations; typically arising from discretizations of partial differential equations that are used to mathematically model various phenomena. The algorithms used to solve these problems are typically iterative in nature, and making efficient use of computational time on High Performance Computing (HPC) clusters involves constantly improving these iterative algorithms. Future HPC platforms are expected to encounter three main problem areas: scalability of code, reliability of hardware, and energy efficiency of the platform. The HPC resources that are expected to run the large programs are planned to consist of billions of processing units that come from more traditional multicore processors as well as a variety of different hardware accelerators. This growth in parallelism leads to the presence of all three problems. Previously, work on algorithm development has focused primarily on creating fault tolerance mechanisms for traditional iterative solvers. Recent work has begun to revisit using asynchronous methods for solving large scale applications, and this dissertation presents research into fault tolerance for fine-grained methods that are asynchronous in nature. Classical convergence results for asynchronous methods are revisited and modified to account for the possible occurrence of a fault, and a variety of techniques for recovery from the effects of a fault are proposed. Examples of how these techniques can be used are shown for various algorithms, including an analysis of a fine-grained algorithm for computing incomplete factorizations. Lastly, numerous modeling and simulation tools for the further construction of iterative algorithms for HPC applications are developed, including numerical models for simulating faults and a simulation framework that can be used to extrapolate the performance of algorithms towards future HPC systems

Efficient fault tolerance for selected scientific computing algorithms on heterogeneous and approximate computer architectures

Scientific computing and simulation technology play an essential role to solve central challenges in science and engineering. The high computational power of heterogeneous computer architectures allows to accelerate applications in these domains, which are often dominated by compute-intensive mathematical tasks. Scientific, economic and political decision processes increasingly rely on such applications and therefore induce a strong demand to compute correct and trustworthy results. However, the continued semiconductor technology scaling increasingly imposes serious threats to the reliability and efficiency of upcoming devices. Different reliability threats can cause crashes or erroneous results without indication. Software-based fault tolerance techniques can protect algorithmic tasks by adding appropriate operations to detect and correct errors at runtime. Major challenges are induced by the runtime overhead of such operations and by rounding errors in floating-point arithmetic that can cause false positives. The end of Dennard scaling induces central challenges to further increase the compute efficiency between semiconductor technology generations. Approximate computing exploits the inherent error resilience of different applications to achieve efficiency gains with respect to, for instance, power, energy, and execution times. However, scientific applications often induce strict accuracy requirements which require careful utilization of approximation techniques. This thesis provides fault tolerance and approximate computing methods that enable the reliable and efficient execution of linear algebra operations and Conjugate Gradient solvers using heterogeneous and approximate computer architectures. The presented fault tolerance techniques detect and correct errors at runtime with low runtime overhead and high error coverage. At the same time, these fault tolerance techniques are exploited to enable the execution of the Conjugate Gradient solvers on approximate hardware by monitoring the underlying error resilience while adjusting the approximation error accordingly. Besides, parameter evaluation and estimation methods are presented that determine the computational efficiency of application executions on approximate hardware. An extensive experimental evaluation shows the efficiency and efficacy of the presented methods with respect to the runtime overhead to detect and correct errors, the error coverage as well as the achieved energy reduction in executing the Conjugate Gradient solvers on approximate hardware

Application of HPC in eddy current electromagnetic problem solution

As engineering problems are becoming more and more advanced, the size of an average model solved by partial differential equations is rapidly growing and, in order to keep simulation times within reasonable bounds, both faster computers and more efficient software implementations are needed. In the first part of this thesis, the full potential of simulation software has been exploited through high performance parallel computing techniques. In particular, the simulation of induction heating processes is accomplished within reasonable solution times, by implementing different parallel direct solvers for large sparse linear system, in the solution process of a commercial software. The performance of such library on shared memory systems has been remarkably improved by implementing a multithreaded version of MUMPS (MUltifrontal Massively Parallel Solver) library, which have been tested on benchmark matrices arising from typical induction heating process simulations. A new multithreading approach and a low rank approximation technique have been implemented and developed by MUMPS team in Lyon and Toulouse. In the context of a collaboration between MUMPS team and DII-University of Padova, a preliminary version of such functionalities could be tested on induction heating benchmark problems, and a substantial reduction of the computational cost and memory requirements could be achieved. In the second part of this thesis, some examples of design methodology by virtual prototyping have been described. Complex multiphysics simulations involving electromagnetic, circuital, thermal and mechanical problems have been performed by exploiting parallel solvers, as developed in the first part of this thesis. Finally, multiobjective stochastic optimization algorithms have been applied to multiphysics 3D model simulations in search of a set of improved induction heating device configurations

Resiliency in numerical algorithm design for extreme scale simulations

This work is based on the seminar titled ‘Resiliency in Numerical Algorithm Design for Extreme Scale Simulations’ held March 1–6, 2020, at Schloss Dagstuhl, that was attended by all the authors. Advanced supercomputing is characterized by very high computation speeds at the cost of involving an enormous amount of resources and costs. A typical large-scale computation running for 48 h on a system consuming 20 MW, as predicted for exascale systems, would consume a million kWh, corresponding to about 100k Euro in energy cost for executing 1023 floating-point operations. It is clearly unacceptable to lose the whole computation if any of the several million parallel processes fails during the execution. Moreover, if a single operation suffers from a bit-flip error, should the whole computation be declared invalid? What about the notion of reproducibility itself: should this core paradigm of science be revised and refined for results that are obtained by large-scale simulation? Naive versions of conventional resilience techniques will not scale to the exascale regime: with a main memory footprint of tens of Petabytes, synchronously writing checkpoint data all the way to background storage at frequent intervals will create intolerable overheads in runtime and energy consumption. Forecasts show that the mean time between failures could be lower than the time to recover from such a checkpoint, so that large calculations at scale might not make any progress if robust alternatives are not investigated. More advanced resilience techniques must be devised. The key may lie in exploiting both advanced system features as well as specific application knowledge. Research will face two essential questions: (1) what are the reliability requirements for a particular computation and (2) how do we best design the algorithms and software to meet these requirements? While the analysis of use cases can help understand the particular reliability requirements, the construction of remedies is currently wide open. One avenue would be to refine and improve on system- or application-level checkpointing and rollback strategies in the case an error is detected. Developers might use fault notification interfaces and flexible runtime systems to respond to node failures in an application-dependent fashion. Novel numerical algorithms or more stochastic computational approaches may be required to meet accuracy requirements in the face of undetectable soft errors. These ideas constituted an essential topic of the seminar. The goal of this Dagstuhl Seminar was to bring together a diverse group of scientists with expertise in exascale computing to discuss novel ways to make applications resilient against detected and undetected faults. In particular, participants explored the role that algorithms and applications play in the holistic approach needed to tackle this challenge. This article gathers a broad range of perspectives on the role of algorithms, applications and systems in achieving resilience for extreme scale simulations. The ultimate goal is to spark novel ideas and encourage the development of concrete solutions for achieving such resilience holistically.Peer Reviewed"Article signat per 36 autors/es: Emmanuel Agullo, Mirco Altenbernd, Hartwig Anzt, Leonardo Bautista-Gomez, Tommaso Benacchio, Luca Bonaventura, Hans-Joachim Bungartz, Sanjay Chatterjee, Florina M. Ciorba, Nathan DeBardeleben, Daniel Drzisga, Sebastian Eibl, Christian Engelmann, Wilfried N. Gansterer, Luc Giraud, Dominik G ̈oddeke, Marco Heisig, Fabienne Jezequel, Nils Kohl, Xiaoye Sherry Li, Romain Lion, Miriam Mehl, Paul Mycek, Michael Obersteiner, Enrique S. Quintana-Ortiz, Francesco Rizzi, Ulrich Rude, Martin Schulz, Fred Fung, Robert Speck, Linda Stals, Keita Teranishi, Samuel Thibault, Dominik Thonnes, Andreas Wagner and Barbara Wohlmuth"Postprint (author's final draft

Methodology for malleable applications on distributed memory systems

A la portada logo BSC(English) The dominant programming approach for scientific and industrial computing on clusters is MPI+X. While there are a variety of approaches within the node, denoted by the ``X'', Message Passing interface (MPI) is the standard for programming multiple nodes with distributed memory. This thesis argues that the OmpSs-2 tasking model can be extended beyond the node to naturally support distributed memory, with three benefits: First, at small to medium scale the tasking model is a simpler and more productive alternative to MPI. It eliminates the need to distribute the data explicitly and convert all dependencies into explicit message passing. It also avoids the complexity of hybrid programming using MPI+X. Second, the ability to offload parts of the computation among the nodes enables the runtime to automatically balance the loads in a full-scale MPI+X program. This approach does not require a cost model, and it is able to transparently balance the computational loads across the whole program, on all its nodes. Third, because the runtime handles all low-level aspects of data distribution and communication, it can change the resource allocation dynamically, in a way that is transparent to the application. This thesis describes the design, development and evaluation of OmpSs-2@Cluster, a programming model and runtime system that extends the OmpSs-2 model to allow a virtually unmodified OmpSs-2 program to run across multiple distributed memory nodes. For well-balanced applications it provides similar performance to MPI+OpenMP on up to 16 nodes, and it improves performance by up to 2x for irregular and unbalanced applications like Cholesky factorization. This work also extended OmpSs-2@Cluster for interoperability with MPI and Barcelona Supercomputing Center (BSC)'s state-of-the-art Dynamic Load Balance (DLB) library in order to dynamically balance MPI+OmpSs-2 applications by transparently offloading tasks among nodes. This approach reduces the execution time of a microscale solid mechanics application by 46% on 64 nodes and on a synthetic benchmark, it is within 10% of perfect load balancing on up to 8 nodes. Finally, the runtime was extended to transparently support malleability for pure OmpSs-2@Cluster programs and interoperate with the Resources Management System (RMS). The only change to the application is to explicitly call an API function to control the addition or removal of nodes. In this regard we additionally provide the runtime with the ability to semi-transparently save and recover part of the application status to perform checkpoint and restart. Such a feature hides the complexity of data redistribution and parallel IO from the user while allowing the program to recover and continue previous executions. Our work is a starting point for future research on fault tolerance. In summary, OmpSs-2@Cluster expands the OmpSs-2 programming model to encompass distributed memory clusters. It allows an existing OmpSs-2 program, with few if any changes, to run across multiple nodes. OmpSs-2@Cluster supports transparent multi-node dynamic load balancing for MPI+OmpSs-2 programs, and enables semi-transparent malleability for OmpSs-2@Cluster programs. The runtime system has a high level of stability and performance, and it opens several avenues for future work.(Español) El modelo de programación dominante para clusters tanto en ciencia como industria es actualmente MPI+X. A pesar de que hay alguna variedad de alternativas para programar dentro de un nodo (indicado por la "X"), el estandar para programar múltiples nodos con memoria distribuida sigue siendo Message Passing Interface (MPI). Esta tesis propone la extensión del modelo de programación basado en tareas OmpSs-2 para su funcionamiento en sistemas de memoria distribuida, destacando 3 beneficios principales: En primer lugar; a pequeña y mediana escala, un modelo basado en tareas es más simple y productivo que MPI y elimina la necesidad de distribuir los datos explícitamente y convertir todas las dependencias en mensajes. Además, evita la complejidad de la programacion híbrida MPI+X. En segundo lugar; la capacidad de enviar partes del cálculo entre los nodos permite a la librería balancear la carga de trabajo en programas MPI+X a gran escala. Este enfoque no necesita un modelo de coste y permite equilibrar cargas transversalmente en todo el programa y todos los nodos. En tercer lugar; teniendo en cuenta que es la librería quien maneja todos los aspectos relacionados con distribución y transferencia de datos, es posible la modificación dinámica y transparente de los recursos que utiliza la aplicación. Esta tesis describe el diseño, desarrollo y evaluación de OmpSs-2@Cluster; un modelo de programación y librería que extiende OmpSs-2 permitiendo la ejecución de programas OmpSs-2 existentes en múltiples nodos sin prácticamente necesidad de modificarlos. Para aplicaciones balanceadas, este modelo proporciona un rendimiento similar a MPI+OpenMP hasta 16 nodos y duplica el rendimiento en aplicaciones irregulares o desbalanceadas como la factorización de Cholesky. Este trabajo incluye la extensión de OmpSs-2@Cluster para interactuar con MPI y la librería de balanceo de carga Dynamic Load Balancing (DLB) desarrollada en el Barcelona Supercomputing Center (BSC). De este modo es posible equilibrar aplicaciones MPI+OmpSs-2 mediante la transferencia transparente de tareas entre nodos. Este enfoque reduce el tiempo de ejecución de una aplicación de mecánica de sólidos a micro-escala en un 46% en 64 nodos; en algunos experimentos hasta 8 nodos se pudo equilibrar perfectamente la carga con una diferencia inferior al 10% del equilibrio perfecto. Finalmente, se implementó otra extensión de la librería para realizar operaciones de maleabilidad en programas OmpSs-2@Cluster e interactuar con el Sistema de Manejo de Recursos (RMS). El único cambio requerido en la aplicación es la llamada explicita a una función de la interfaz que controla la adición o eliminación de nodos. Además, se agregó la funcionalidad de guardar y recuperar parte del estado de la aplicación de forma semitransparente con el objetivo de realizar operaciones de salva-reinicio. Dicha funcionalidad oculta al usuario la complejidad de la redistribución de datos y las operaciones de lectura-escritura en paralelo, mientras permite al programa recuperar y continuar ejecuciones previas. Este es un punto de partida para futuras investigaciones en tolerancia a fallos. En resumen, OmpSs-2@Cluster amplía el modelo de programación de OmpSs-2 para abarcar sistemas de memoria distribuida. El modelo permite la ejecución de programas OmpSs-2 en múltiples nodos prácticamente sin necesidad de modificarlos. OmpSs-2@Cluster permite además el balanceo dinámico de carga en aplicaciones híbridas MPI+OmpSs-2 ejecutadas en varios nodos y es capaz de realizar maleabilidad semi-transparente en programas OmpSs-2@Cluster puros. La librería tiene un niveles de rendimiento y estabilidad altos y abre varios caminos para trabajos futuro.Arquitectura de computador

Software for Exascale Computing - SPPEXA 2016-2019

This open access book summarizes the research done and results obtained in the second funding phase of the Priority Program 1648 "Software for Exascale Computing" (SPPEXA) of the German Research Foundation (DFG) presented at the SPPEXA Symposium in Dresden during October 21-23, 2019. In that respect, it both represents a continuation of Vol. 113 in Springer’s series Lecture Notes in Computational Science and Engineering, the corresponding report of SPPEXA’s first funding phase, and provides an overview of SPPEXA’s contributions towards exascale computing in today's sumpercomputer technology. The individual chapters address one or more of the research directions (1) computational algorithms, (2) system software, (3) application software, (4) data management and exploration, (5) programming, and (6) software tools. The book has an interdisciplinary appeal: scholars from computational sub-fields in computer science, mathematics, physics, or engineering will find it of particular interest

Reducing Communication in the Solution of Linear Systems

There is a growing performance gap between computation and communication on modern computers, making it crucial to develop algorithms with lower latency and bandwidth requirements. Because systems of linear equations are important for numerous scientific and engineering applications, I have studied several approaches for reducing communication in those problems. First, I developed optimizations to dense LU with partial pivoting, which downstream applications can adopt with little to no effort. Second, I consider two techniques to completely replace pivoting in dense LU, which can provide significantly higher speedups, albeit without the same numerical guarantees as partial pivoting. One technique uses randomized preprocessing, while the other is a novel combination of block factorization and additive perturbation. Finally, I investigate using mixed precision in GMRES for solving sparse systems, which reduces the volume of data movement, and thus, the pressure on the memory bandwidth

ParallelLCA : a foreground aware parallel calculator for life cycle assessment

Life Cycle Assessment (LCA), which aims to assess the environmental impacts during the life cycle of a system product (S) (e.g., production of aluminum in Quebec), can be used to compare different systems built with different types of materials to determine which is the least harmful to the environment. The calculation in LCA represents a computational challenge as it is dependent on the size of the system, the number of iterations in the Monte-Carlo simulation, and the number of uncertain variables in the system. First, the solving of a linear system of dimensions in the order of 10,000 equations by 10,000 unknown variables is required for the base case. Second, the building of a graph iterative in nature with minimum dimensions of 10,000 vertices. Third, the computing of a Monte-Carlo simulation requiring several thousands of iterations to converge is to be computed. Finally, a sensitivity analysis which requires the computing of millions of correlations between vectors each having a dimension that is proportional to the number of iterations in the Monte-Carlo simulation. To best solve the computational challenges present in LCA, this research benefits from well established libraries that solve large sparse linear systems and performs large sparse matrix computing. Also, this thesis adopted mathematical optimizations that removed the matrix inverse step from the contribution analysis module, which is very expensive, as well as other algorithmic optimizations that removed the large and variant part of the LCA supply-chain from the matrix component of the various calculation phases. Furthermore, this research experimented with libraries such as OpenMP, MPI, and Apache Spark to parallelize the computation. First, the thesis will discuss the literature regarding these computational opportunities. Second, it will present a proposed LCA calculator for implementing an efficient LCA computation. Finally, it will present the performance of computing the different phases of LCA for various dimensions of the system (S) and concludes with suggestions for improvement and future development