While many of the architectural details of future exascale-class high performance computer systems are still a matter of intense research, there appears to be a general consensus that they will be strongly heterogeneous, featuring “standard” as well as “accelerated” resources. Today, such resources are available as multicore processors, graphics processing units (GPUs), and other accelerators such as the Intel Xeon Phi. Any software infrastructure that claims usefulness for such environments must be able to meet their inherent challenges: massive multi-level parallelism, topology, asynchronicity, and abstraction. The “General, Hybrid, and Optimized Sparse Toolkit” (GHOST) is a collection of building blocks that targets algorithms dealing with sparse matrix representations on current and future large-scale systems. It implements the “MPI+X” paradigm, has a pure C interface, and provides hybrid-parallel numerical kernels, intelligent resource management, and truly heterogeneous parallelism for multicore CPUs, Nvidia GPUs, and the Intel Xeon Phi. We describe the details of its design with respect to the challenges posed by modern heterogeneous supercomputers and recent algorithmic developments. Implementation details which are indispensable for achieving high efficiency are pointed out and their necessity is justified by performance measurements or predictions based on performance models. We also provide instructions on how to make use of GHOST in existing software packages, together with a case study which demonstrates the applicability and performance of GHOST as a component within a larger software stack. The library code and several applications are available as open source.

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The ESSEX project is an ongoing effort to provide exascale-enabled sparse eigensolvers, especially for quantum physics and related application areas. In this paper we first briefly summarize some key achievements that have been made within this project. Then we focus on a projection-based eigensolver with polynomial approximation of the projector. This eigensolver can be used for computing hundreds of interior eigenvalues of large sparse matrices. We describe techniques that allow using lower-degree polynomials than possible with standard Chebyshev expansion of the window function and kernel smoothing. With these polynomials, the degree, and thus the number of matrix–vector multiplications, typically can be reduced by roughly one half, resulting in comparable savings in runtime.

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As we approach the exascale computing era, disruptive changes in the software landscape are required to tackle the challenges posed by manycore CPUs and accelerators. We discuss the development of a new ‘exascale enabled’ sparse solver repository (the ESSR) that addresses these challenges—from fundamental design considerations and development processes to actual implementations of some prototypical iterative schemes for computing eigenvalues of sparse matrices. Key features of the ESSR include holistic performance engineering, tight integration between software layers and mechanisms to mitigate hardware failures.

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Numerous challenges have to be mastered as applications in scientific computing are being developed for post-petascale parallel systems. While ample parallelism is usually available in the numerical problems at hand, the efficient use of supercomputer resources requires not only good scalability but also a verifiably effective use of resources on the core, the processor, and the accelerator level. Furthermore, power dissipation and energy consumption are becoming further optimization targets besides time-to-solution. Performance Engineering (PE) is the pivotal strategy for developing effective parallel code on all levels of modern architectures. In this paper we report on the development and use of low-level parallel building blocks in the GHOST library (“General, Hybrid, and Optimized Sparse Toolkit”). We demonstrate the use of PE in optimizing a density of states computation using the Kernel Polynomial Method, and show that reduction of runtime and reduction of energy are literally the same goal in this case. We also give a brief overview of the capabilities of GHOST and the applications in which it is being used successfully.

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Block variants of the Jacobi-Davidson method for computing a few eigenpairs of a large sparse matrix are known to improve the robustness of the standard algorithm when it comes to computing multiple or clustered eigenvalues. In practice, however, they are typically avoided because the total number of matrix-vector operations increases. In this paper we present the implementation of a block Jacobi-Davidson solver. By detailed performance engineering and numerical experiments we demonstrate that the increase in operations is typically more than compensated by performance gains through better cache usage on modern CPUs, resulting in a method that is both more efficient and robust than its single vector counterpart. The steps to be taken to achieve a block speedup involve both kernel optimizations for sparse matrix and block vector operations, and algorithmic choices to allow using blocked operations in most parts of the computation. We discuss the aspect of avoiding synchronization in the algorithm and show by numerical experiments with our hybrid parallel implementation that a significant speedup through blocking can be achieved for a variety of matrices on up to 5 120 CPU cores as long as at least about 20 eigenpairs are sought.@en

The ESSEX project investigates computational issues arising at exascale for large-scale sparse eigenvalue problems and develops programming concepts and numerical methods for their solution. The project pursues a coherent co-design of all software layers where a holistic performance engineering process guides code development across the classic boundaries of application, numerical method, and basic kernel library. Within ESSEX the numerical methods cover widely applicable solvers such as classic Krylov, Jacobi-Davidson, or the recent FEAST methods, as well as domain-specific iterative schemes relevant for the ESSEX quantum physics application. This report introduces the project structure and presents selected results which demonstrate the potential impact of ESSEX for efficient sparse solvers on highly scalable heterogeneous supercomputers.

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