Improving Performance of Iterative Methods by Lossy Checkponting

Iterative methods are commonly used approaches to solve large, sparse linear systems, which are fundamental operations for many modern scientific simulations. When the large-scale iterative methods are running with a large number of ranks in parallel, they have to checkpoint the dynamic variables periodically in case of unavoidable fail-stop errors, requiring fast I/O systems and large storage space. To this end, significantly reducing the checkpointing overhead is critical to improving the overall performance of iterative methods. Our contribution is fourfold. (1) We propose a novel lossy checkpointing scheme that can significantly improve the checkpointing performance of iterative methods by leveraging lossy compressors. (2) We formulate a lossy checkpointing performance model and derive theoretically an upper bound for the extra number of iterations caused by the distortion of data in lossy checkpoints, in order to guarantee the performance improvement under the lossy checkpointing scheme. (3) We analyze the impact of lossy checkpointing (i.e., extra number of iterations caused by lossy checkpointing files) for multiple types of iterative methods. (4)We evaluate the lossy checkpointing scheme with optimal checkpointing intervals on a high-performance computing environment with 2,048 cores, using a well-known scientific computation package PETSc and a state-of-the-art checkpoint/restart toolkit. Experiments show that our optimized lossy checkpointing scheme can significantly reduce the fault tolerance overhead for iterative methods by 23%~70% compared with traditional checkpointing and 20%~58% compared with lossless-compressed checkpointing, in the presence of system failures.Comment: 14 pages, 10 figures, HPDC'1

Reliability for exascale computing : system modelling and error mitigation for task-parallel HPC applications

As high performance computing (HPC) systems continue to grow, their fault rate increases. Applications running on these systems have to deal with rates on the order of hours or days. Furthermore, some studies for future Exascale systems predict the rates to be on the order of minutes. As a result, efficient fault tolerance solutions are needed to be able to tolerate frequent failures. A fault tolerance solution for future HPC and Exascale systems must be low-cost, efficient and highly scalable. It should have low overhead in fault-free execution and provide fast restart because long-running applications are expected to experience many faults during the execution. Meanwhile task-based dataflow parallel programming models (PM) are becoming a popular paradigm in HPC applications at large scale. For instance, we see the adaptation of task-based dataflow parallelism in OpenMP 4.0, OmpSs PM, Argobots and Intel Threading Building Blocks. In this thesis we propose fault-tolerance solutions for task-parallel dataflow HPC applications. Specifically, first we design and implement a checkpoint/restart and message-logging framework to recover from errors. We then develop performance models to investigate the benefits of our task-level frameworks when integrated with system-wide checkpointing. Moreover, we design and implement selective task replication mechanisms to detect and recover from silent data corruptions in task-parallel dataflow HPC applications. Finally, we introduce a runtime-based coding scheme to detect and recover from memory errors in these applications. Considering the span of all of our schemes, we see that they provide a fairly high failure coverage where both computation and memory is protected against errors.A medida que los Sistemas de Cómputo de Alto rendimiento (HPC por sus siglas en inglés) siguen creciendo, también las tasas de fallos aumentan. Las aplicaciones que se ejecutan en estos sistemas tienen una tasa de fallos que pueden estar en el orden de horas o días. Además, algunos estudios predicen que los fallos estarán en el orden de minutos en los Sistemas Exascale. Por lo tanto, son necesarias soluciones eficientes para la tolerancia a fallos que puedan tolerar fallos frecuentes. Las soluciones para tolerancia a fallos en los Sistemas futuros de HPC y Exascale tienen que ser de bajo costo, eficientes y altamente escalable. El sobrecosto en la ejecución sin fallos debe ser bajo y también se debe proporcionar reinicio rápido, ya que se espera que las aplicaciones de larga duración experimenten muchos fallos durante la ejecución. Por otra parte, los modelos de programación paralelas basados en tareas ordenadas de acuerdo a sus dependencias de datos, se están convirtiendo en un paradigma popular en aplicaciones HPC a gran escala. Por ejemplo, los siguientes modelos de programación paralela incluyen este tipo de modelo de programación OpenMP 4.0, OmpSs, Argobots e Intel Threading Building Blocks. En esta tesis proponemos soluciones de tolerancia a fallos para aplicaciones de HPC programadas en un modelo de programación paralelo basado tareas. Específicamente, en primer lugar, diseñamos e implementamos mecanismos “checkpoint/restart” y “message-logging” para recuperarse de los errores. Para investigar los beneficios de nuestras herramientas a nivel de tarea cuando se integra con los “system-wide checkpointing” se han desarrollado modelos de rendimiento. Por otra parte, diseñamos e implementamos mecanismos de replicación selectiva de tareas que permiten detectar y recuperarse de daños de datos silenciosos en aplicaciones programadas siguiendo el modelo de programación paralela basadas en tareas. Por último, se introduce un esquema de codificación que funciona en tiempo de ejecución para detectar y recuperarse de los errores de la memoria en estas aplicaciones. Todos los esquemas propuestos, en conjunto, proporcionan una cobertura bastante alta a los fallos tanto si estos se producen el cálculo o en la memoria.Postprint (published version

Algorithm-based Fault Tolerance for Dense Matrix Factorizations, Multiple Failures and Accuracy

Dense matrix factorizations, such as LU, Cholesky and QR, are widely used for scientific applications that require solving systems of linear equations, eigenvalues and linear least squares problems. Such computations are normally carried out on supercomputers, whose ever-growing scale induces a fast decline of the Mean Time To Failure (MTTF). This article proposes a new hybrid approach, based on Algorithm-Based Fault Tolerance (ABFT), to help matrix factorizations algorithms survive fail-stop failures. We consider extreme conditions, such as the absence of any reliable node and the possibility of losing both data and checksum from a single failure. We will present a generic solution for protecting the right factor, where the updates are applied, of all above mentioned factorizations. For the left factor, where the panel has been applied, we propose a scalable checkpointing algorithm. This algorithm features high degree of checkpointing parallelism and cooperatively utilizes the checksum storage leftover from the right factor protection. The fault-tolerant algorithms derived from this hybrid solution is applicable to a wide range of dense matrix factorizations, with minor modifications. Theoretical analysis shows that the fault tolerance overhead decreases inversely to the scaling in the number of computing units and the problem size. Experimental results of LU and QR factorization on the Kraken (Cray XT5) supercomputer validate the theoretical evaluation and confirm negligible overhead, with- and without-errors. Applicability to tolerate multiple failures and accuracy after multiple recovery is also considered.</jats:p

Algorithm-Based Fault Tolerance for Two-Sided Dense Matrix Factorizations

The mean time between failure (MTBF) of large supercomputers is decreasing, and future exascale computers are expected to have a MTBF of around 30 minutes. Therefore, it is urgent to prepare important algorithms for future machines with such a short MTBF. Eigenvalue problems (EVP) and singular value problems (SVP) are common in engineering and scientific research. Solving EVP and SVP numerically involves two-sided matrix factorizations: the Hessenberg reduction, the tridiagonal reduction, and the bidiagonal reduction. These three factorizations are computation intensive, and have long running times. They are prone to suffer from computer failures. We designed algorithm-based fault tolerant (ABFT) algorithms for the parallel Hessenberg reduction and the parallel tridiagonal reduction. The ABFT algorithms target fail-stop errors. These two fault tolerant algorithms use a combination of ABFT and diskless checkpointing. ABFT is used to protect frequently modified data . We carefully design the ABFT algorithm so the checksums are valid at the end of each iterative cycle. Diskless checkpointing is used for rarely modified data. These checkpoints are in the form of checksums, which are small in size, so the time and storage cost to store them in main memory is small. Also, there are intermediate results which need to be protected for a short time window. We store a copy of this data on the neighboring process in the process grid. We also designed algorithm-based fault tolerant algorithms for the CPU-GPU hybrid Hessenberg reduction algorithm and the CPU-GPU hybrid bidiagonal reduction algorithm. These two fault tolerant algorithms target silent errors. Our design employs both ABFT and diskless checkpointing to provide data redundancy. The low cost error detection uses two dot products and an equality test. The recovery protocol uses reverse computation to roll back the state of the matrix to a point where it is easy to locate and correct errors. We provided theoretical analysis and experimental verification on the correctness and efficiency of our fault tolerant algorithm design. We also provided mathematical proof on the numerical stability of the factorization results after fault recovery. Experimental results corroborate with the mathematical proof that the impact is mild

Hard and Soft Error Resilience for One-sided Dense Linear Algebra Algorithms

Dense matrix factorizations, such as LU, Cholesky and QR, are widely used by scientific applications that require solving systems of linear equations, eigenvalues and linear least squares problems. Such computations are normally carried out on supercomputers, whose ever-growing scale induces a fast decline of the Mean Time To Failure (MTTF). This dissertation develops fault tolerance algorithms for one-sided dense matrix factorizations, which handles Both hard and soft errors. For hard errors, we propose methods based on diskless checkpointing and Algorithm Based Fault Tolerance (ABFT) to provide full matrix protection, including the left and right factor that are normally seen in dense matrix factorizations. A horizontal parallel diskless checkpointing scheme is devised to maintain the checkpoint data with scalable performance and low space overhead, while the ABFT checksum that is generated before the factorization constantly updates itself by the factorization operations to protect the right factor. In addition, without an available fault tolerant MPI supporting environment, we have also integrated the Checkpoint-on-Failure(CoF) mechanism into one-sided dense linear operations such as QR factorization to recover the running stack of the failed MPI process. Soft error is more challenging because of the silent data corruption, which leads to a large area of erroneous data due to error propagation. Full matrix protection is developed where the left factor is protected by column-wise local diskless checkpointing, and the right factor is protected by a combination of a floating point weighted checksum scheme and soft error modeling technique. To allow practical use on large scale system, we have also developed a complexity reduction scheme such that correct computing results can be recovered with low performance overhead. Experiment results on large scale cluster system and multicore+GPGPU hybrid system have confirmed that our hard and soft error fault tolerance algorithms exhibit the expected error correcting capability, low space and performance overhead and compatibility with double precision floating point operation

Fault Tolerant and Energy Efficient One-Sided Matrix Decompositions on Heterogeneous Systems with GPUs

Heterogeneous computing system with both CPUs and GPUs has become a class of widely used hardware architecture in supercomputers. As heterogeneous systems delivering higher computational performance, they are being built with an increasing number of complex components. This is anticipated that these systems will be more susceptible to hardware faults with higher power consumption. Numerical linear algebra libraries are used in a wide spectrum of high-performance scientific applications. Among numerical linear algebra operations, one-sided matrix decompositions can sometimes take a large portion of execution time or even dominate the whole scientific application execution. Due to the computational characteristic of one-sided matrix decompositions, they are very suitable for computation platforms such as heterogeneous systems with CPUs and GPUs. Many works have been done to implement and optimize one-sided matrix decompositions on heterogeneous systems with CPUs and GPUs. However, it is challenging to enable stable and high performance one-sided matrix decompositions running on computing platforms that are unreliable and high energy consumption. So, in this thesis, we aim to develop novel fault tolerance and energy efficiency optimizations for one-sided matrix decompositions on heterogeneous systems with CPUs and GPUs.To improve reliability and energy efficiency, extensive researches have been done on developing and optimizing fault tolerance methods and energy-saving strategies for one-sided matrix decompositions. However, current designs still have several limitations: (1) Little has been done on developing and optimizing fault tolerance method for one-sided matrix decompositions on heterogeneous systems with GPUs; (2) Limited by the protection coverage and strength, existing fault tolerance works provide insufficient protection when applied to one-sided matrix decompositions on heterogeneous systems with GPUs; (3) Lack the knowledge of algorithms, existing system level energy saving solutions cannot achieve the optimal energy savings due to potentially inaccurate and high-cost workload prediction they rely on when they are used in one-sided matrix decompositions; (4) It is challenging to apply both fault tolerance techniques and energy saving strategies to one-side matrix decompositions at the same time given that their current designs are not naturally compatible with each other.To address the first problem, based on the original (Algorithm Based Fault Tolerance) ABFT, we develop the first ABFT for matrix decomposition on heterogeneous systems with GPUs together with the novel storage errors protection and several optimization techniques specifically for GPUs. As for the second problem, we design a novel checksum scheme for ABFT that allows data stored in matrices to be encoded in two dimensions. This stronger checksum encoding mechanism enables much stronger protection including enhanced error propagation protection. In addition, we introduce a more efficient checking scheme. By prioritizing the checksum verification according to the sensitivity of matrix operations to soft errors with optimized checksum verification kernel for GPUs, we can achieve strong protect to matrix decompositions with comparable overhead. For the third problem, to improve energy efficiency for one-sided matrix decompositions, we introduce an algorithm-based energy-saving approach designed to maximize energy savings by utilizing algorithmic characteristics. Our approach can predict program execution behavior much more accurately, which is difficult for system level solutions for applications with variable execution characteristics. Experiments show that our approach can lead to much higher energy saving than existing works. Finally, for the fourth problem, we propose a novel energy saving approach for one-sided matrix decompositions on heterogeneous systems with GPUs. It allows energy saving strategies and fault tolerance techniques to be enabled at the same time without brings performance impact or extra energy cost

Extensions of Task-based Runtime for High Performance Dense Linear Algebra Applications

On the road to exascale computing, the gap between hardware peak performance and application performance is increasing as system scale, chip density and inherent complexity of modern supercomputers are expanding. Even if we put aside the difficulty to express algorithmic parallelism and to efficiently execute applications at large scale, other open questions remain. The ever-growing scale of modern supercomputers induces a fast decline of the Mean Time To Failure. A generic, low-overhead, resilient extension becomes a desired aptitude for any programming paradigm. This dissertation addresses these two critical issues, designing an efficient unified linear algebra development environment using a task-based runtime, and extending a task-based runtime with fault tolerant capabilities to build a generic framework providing both soft and hard error resilience to task-based programming paradigm. To bridge the gap between hardware peak performance and application perfor- mance, a unified programming model is designed to take advantage of a lightweight task-based runtime to manage the resource-specific workload, and to control the data ow and parallel execution of tasks. Under this unified development, linear algebra tasks are abstracted across different underlying heterogeneous resources, including multicore CPUs, GPUs and Intel Xeon Phi coprocessors. Performance portability is guaranteed and this programming model is adapted to a wide range of accelerators, supporting both shared and distributed-memory environments. To solve the resilient challenges on large scale systems, fault tolerant mechanisms are designed for a task-based runtime to protect applications against both soft and hard errors. For soft errors, three additions to a task-based runtime are explored. The first recovers the application by re-executing minimum number of tasks, the second logs intermediary data between tasks to minimize the necessary re-execution, while the last one takes advantage of algorithmic properties to recover the data without re- execution. For hard errors, we propose two generic approaches, which augment the data logging mechanism for soft errors. The first utilizes non-volatile storage device to save logged data, while the second saves local logged data on a remote node to protect against node failure. Experimental results have confirmed that our soft and hard error fault tolerant mechanisms exhibit the expected correctness and efficiency