The performance of most scientific applications depends on efficient mathematical libraries. For example, scientific applications like the plane wave based Density Functional Theory approach for electronic structure calculations uses highly optimized libraries for Fourier transforms, dense linear algebra (orthogonalization) and sparse linear algebra (non-local projectors in real space). Although vendor-tuned libraries offer efficient implementations for each standalone mathematical kernel, the partitioning of those calls into sequentially invoked kernels inhibits cross-kernel optimizations that could improve data locality across memory bound operations. In this work we show that, by expressing these kernels as an operation on high dimensional tensors, cross-kernel dataflow optimizations that span FFT, dense and sparse linear algebra, can be readily exposed and exploited. We outline a systematic way of merging the Fourier transforms with the linear algebra computations, improving data locality and reducing data movement to main memory. We show that compared to conventional implementations, this streaming/dataflow approach offers 2x speedup on GPUs and 8x/12x speedup on CPUs compared to a baseline code that uses vendor-optimized libraries. Although we use Density Functional Theory to demonstrate the value of our approach, our methodology is broadly applicable to other applications that use Fourier transforms and linear algebra operations as building blocks.

This paper reports on the performance of a preconditioned conjugate gradient based iterative eigensolver using an unconstrained energy functional minimization scheme. In contrast to standard implementations, this scheme avoids an explicit reorthogonalization of the trial eigenvectors and becomes an attractive alternative for the solution of very large problems. The unconstrained formulation is implemented in the first-principles materials and chemistry CP2K code, which performs electronic structure calculations based on a density functional theory approximation to the solution of the many-body Schrödinger equation. We study the convergence of the unconstrained formulation, as well as its parallel scaling, on a Cray XC40 at the National Energy Research Scientific Computing Center (NERSC). The systems we use in our studies are bulk liquid water, a supramolecular catalyst gold(III)-complex, a bilayer of MoS2-WSe2 and a divacancy point defect in silicon, with the number of atoms ranging from 2,247 to 12,288. We show that the unconstrained formulation with an appropriate preconditioner has good convergence properties and scales well to 230k cores, roughly 38% of the full machine.

After a decade where HEC (high-end computing) capability was dominated by the rapid pace of improvements to CPU clock frequency, the performance of next-generation supercomputers is increasingly differentiated by varying interconnect designs and levels of integration. Understanding the tradeoffs of these system designs, in the context of high-end numerical simulations, is a key step towards making effective petascale computing a reality. This work represents one of the most comprehensive performance evaluation studies to date on modern NEC systems, including the IBM Power5, AMD Opteron, IBM BG/L, and Cray X1E. A novel aspect of our study is the emphasis on full applications, with real input data at the scale desired by computational scientists in their unique domain. We examine six candidate ultra-scale applications, representing a broad range of algorithms and computational structures. Our work includes the highest concurrency experiments to date on five of our six applications, including 32K processor scalability for two of our codes and describe several successful optimizations strategies on BG/L, as well as improved X1E vectorization. Overall results indicate that our evaluated codes have the potential to effectively utilize petascale resources; however, several applications would require reengineering to incorporate the additional levels of parallelism necessary to achieve the vast concurrency of upcoming ultra-scale systems.

The last decade has witnessed a rapid proliferation of superscalar cache-based microprocessors to build high-end computing (HEC) platforms, primarily because of their generality, scalability, and cost effectiveness. However, the growing gap between sustained and peak performance for full-scale scientific applications on conventional supercomputers has become a major concern in high performance computing, requiring significantly larger systems and application scalability than implied by peak performance in order to achieve desired performance. The latest generation of custom-built parallel vector systems have the potential to address this issue for numerical algorithms with sufficient regularity in their computational structure. In this work we explore applications drawn from four areas: atmospheric modeling (CAM), magnetic fusion (GTC), plasma physics (LBMHD3D), and material science (PARATEC). We compare performance of the vector-based Cray X1, Earth Simulator, and newly-released NEC SX-8 and Cray X1E, with performance of three leading commodity-based superscalar platforms utilizing the IBM Power3, Intel Itanium2, and AMD Opteron processors. Our work makes several significant contributions: the first reported vector performance results for CAM simulations utilizing a finite-volume dynamical core on a high-resolution atmospheric grid; a new data-decomposition scheme for GTC that (for the first time) enables a breakthrough of the Teraflop barrier; the introduction of a new three-dimensional Lattice Boltzmann magneto-hydrodynamic implementation used to study the onset evolution of plasma turbulence that achieves over 26Tflop/s on 4800 ES promodity-based superscalar platforms utilizing the IBM Power3, Intel Itanium2, and AMD Opteron processors, with modern parallel vector systems: the Cray X1, Earth Simulator (ES), and the NEC SX-8. Additionally, we examine performance of CAM on the recently-released Cray X1E. Our research team was the first international group to conduct a performance evaluation study at the Earth Simulator Center; remote ES access is not available. Our work builds on our previous efforts [16, 17] and makes several significant contributions: the first reported vector performance results for CAM simulations utilizing a finite-volume dynamical core on a high-resolution atmospheric grid; a new datadecomposition scheme for GTC that (for the first time) enables a breakthrough of the Teraflop barrier; the introduction of a new three-dimensional Lattice Boltzmann magneto-hydrodynamic implementation used to study the onset evolution of plasma turbulence that achieves over 26Tflop/s on 4800 ES processors; and the largest PARATEC cell size atomistic simulation to date. Overall, results show that the vector architectures attain unprecedented aggregate performance across our application suite, demonstrating the tremendous potential of modern parallel vector systems.

The growing gap between sustained and peak performance for scientific applications is a well-known problem in high-performance computing. The recent development of parallel vector systems offers the potential to reduce this gap for many computational science codes and deliver a substantial increase in computing capabilities. This paper examines the intranode performance of the NEC SX-6 vector processor, and compares it against the cache-based IBM Power3 and Power4 superscalar architectures, across a number of key scientific computing areas. First, we present the performance of a microbenchmark suite that examines many low-level machine characteristics. Next, we study the behavior of the NAS Parallel Benchmarks. Finally, we evaluate the performance of several scientific computing codes. Overall results demonstrate that the SX-6 achieves high performance on a large fraction of our application suite and often significantly outperforms the cache-based architectures. However, certain classes of applications are not easily amenable to vectorization and would require extensive algorithm and implementation reengineering to utilize the SX-6 effectively. Copyright © 2005 John Wiley & Sons, Ltd.

Computational scientists have seen a frustrating trend of stagnating application performance despite dramatic increases in the claimed peak capability of high performance computing systems. This trend has been widely attributed to the use of superscalar-based commodity components who’s architectural designs offer a balance between memory performance, network capability, and execution rate that is poorly matched to the requirements of large-scale numerical computations. Recently, two innovative parallel-vector architectures have become operational: the Japanese Earth Simulator (ES) and the Cray X1. In order to quantify what these modern vector capabilities entail for the scientists that rely on modeling and simulation, it is critical to evaluate this architectural paradigm in the context of demanding computational algorithms. Our evaluation study examines four diverse scientific applications with the potential to run at ultrascale, from the areas of plasma physics, material science, astrophysics, and magnetic fusion. We compare performance between the vector-based ES and X1, with leading superscalar-based platforms: the IBM Power3/4 and the SGI Altix. Our research team was the first international group to conduct a performance evaluation study at the Earth Simulator Center; remote ES access in not available. Results demonstrate that the vector systems achieve excellent performance on our application suite - the highest of any architecture tested to date. However, vectorization of a particle-in-cell code highlights the potential difficulty of expressing irregularly structured algorithms as data-parallel programs.

The growing gap between sustained and peak performance for scientific applications is a well-known problem in high end computing. The recent development of parallel vector systems offers the potential to bridge this gap for many computational science codes and deliver a substantial increase in comput-ing capabilities. This paper examines the intranode performance of the NEC SX-6 vector processor and the cache-based IBM Power3/4 superscalar architectures across a number of scientific computing areas. First, we present the performance of a microbenchmark suite that examines low-level machine characteristics. Next, we study the behavior of the NAS Parallel Benchmarks. Finally, we evaluate the performance of several scientific computing codes. Results demonstrate that the SX-6 achieves high performance on a large fraction of our applications and often significantly outperforms the cache-based architectures. However, certain applications are not easily amenable to vectorization and would require extensive algorithm and implementation reengineering to utilize the SX-6 effectively.

In the heart of most electronic structure simulation programs, there is a routine to find the solution of eigenvalue problems. Solving these eigenvalue problems usually dominates the computer time used for the whole simulation[6]. Because of their physical properties, these eigenvalue problems are always symmetric real or Hermitian. The dimensions of the matrices involved are usually very large and a large number of eigenvalues and their corresponding eigenvectors are needed to compute the desired physical quantities. To solve this type of problems, we introduce a variant of the Lanczos method called the thick-restart Lanczos method. In material science, this method is most appropriate for non-selfconsistent cases where the eigenvalue problems are linear and the number of required eigenvalues is relatively small compared to the size of the matrix.The Lanczos method is very simple and yet effective in finding eigenvalues. It is also well suited for parallel computing. There are two common ways of implementing the Lanczos method depending on whether the Lanczos vectors are stored. When the Lanczos vectors are not stored, they may lose orthogonality and the Lanczos method may generate spurious eigenvalues [2, 10]. Though spurious eigenvalues can be effectively identified, we still prefer not to deal with the spurious eigenvalues. When the Lanczos vectors are stored, the loss of orthogonality problem can be corrected by re-orthogonalization [4, 5, 7]. No spurious eigenvalue is generated in this case. However, because each Lanczos step generates one vector, a large amount of computer memory may be required to store all the Lanczos vectors. To limit the maximum amount of memory used, we typically restart the Lanczos algorithm after a certain number of steps. The restarted versions usually use considerably more matrix-vector multiplications than the non-restarted version. In recent years, newly developed restarting strategies have significantly reduced the number of matrix-vector multiplications used. The two most successful ones are the implicit restarting technique [1, 3, 8] and the dynamic thick-restart technique [9, 12]. For symmetric or Hermitian eigenvalue problems, these two schemes are equivalent. Because the thick-restart scheme is easier to implement and it is slightly more flexible than the implicit restarted scheme [9, 12], the new method described here uses the thick-restart scheme. Other thick-restart eigenvalue methods, e.g., the thick-restart Davidson method, can be applied on symmetric eigenvalue problems as well. Compared to them, the main advantage of the new scheme is that it uses less arithmetic operations by taking full advantage of the symmetry of the matrix [13].