The tridiagonal solver is an important kernel in many scientific and engineering applications. Although quite a few parallel algorithms have been exploited recently, challenges still remain when solving tridiagonal systems on many-core architectures. In this paper, quantitative analysis is conducted to guide the selection of algorithms on different architectures, and a fast register-PCR-pThomas algorithm for Intel MIC architecture is presented to achieve the best utilization of in-core vectorization and registers. By choosing the most suitable number of PCR reductions, we further propose an improved algorithm, named register-PCR-half-pThomas algorithm, which minimizes the computational cost and the number of registers for use. The optimizations of manual prefetching and strength reduction are also discussed for the new algorithm. Evaluation on Intel Xeon Phi 7110P shows that our register-PCR-pThomas can outperform the MKL solver by 7.7 times in single precision and 3.4 times in double precision. Moreover, our optimized register-PCR-half-pThomas can further boost the performance by another 43.9% in single precision and 20.4% in double precision. Our best algorithm can outperform the latest official GPU library (cuSPARSE) on Nvidia K40 by 3.7 times and 2.4 times in single and double precision, respectively.

In this work an ultra-scalable algorithm is designed and optimized to accelerate a 3D compressible Euler atmospheric model on the CPU-MIC hybrid system of Tianhe-2. We first reformulate the mesocale model to avoid long-latency operations, and then employ carefully designed inter-node and intra-node domain decomposition algorithms to achieve balance utilization of different computing units. Proper communication-computation overlap and concurrent data transfer methods are utilized to reduce the cost of data movement at scale. A variety of optimization techniques on both the CPU side and the accelerator side are exploited to enhance the in-socket performance. The proposed hybrid algorithm successfully scales to 6,144 Tianhe-2 nodes with a nearly ideal weak scaling efficiency, and achieve over 8 percent of the peak performance in double precision. This ultra-scalable hybrid algorithm may be of interest to the community to accelerating atmospheric models on increasingly dominated heterogeneous supercomputers.

Stencils are among the most important and time-consuming kernels in many applications. While stencil optimization has been a well-studied topic on CPU platforms, achieving higher performance and efficiency for the evolving numerical stencils on the more recent multi-core and many-core architectures is still an important issue. In this paper, we explore a number of different stencils, ranging from a basic 7-point Jacobi stencil to more complex high-order stencils used in finer numerical simulations. By optimizing and analyzing those stencils on the latest multi-core and many-core architectures (the Intel Sandy Bridge processor, the Intel Xeon Phi coprocessor, and the NVIDIA Fermi C2070 and Kepler K20x GPUs), we investigate the algorithmic and architectural factors that determine the performance and efficiency of the resulting designs. While multi-threading, vectorization, and optimization on cache and other fast buffers are still the most important techniques that provide performance, we observe that the different memory hierarchy and the different mechanism for issuing and executing parallel instructions lead to the different performance behaviors on CPU, MIC and GPU. With vector-like processing units becoming the major provider of computing power on almost all architectures, the compiler's inability to align all the computing and memory operations would become the major bottleneck from getting a high efficiency on current and future platforms. Our specific optimization of the complex WNAD stencil on GPU provides a good example of what the compiler could do to help.

This paper presents a hybrid algorithm for the petascale global simulation of atmospheric dynamics on Tianhe-2, the world's current top-ranked supercomputer developed by China's National University of Defense Technology (NUDT). Tianhe-2 is equipped with both Intel Xeon CPUs and Intel Xeon Phi accelerators. A key idea of the hybrid algorithm is to enable flexible domain partition between an arbitrary number of processors and accelerators, so as to achieve a balanced and efficient utilization of the entire system. We also present an asynchronous and concurrent data transfer scheme to reduce the communication overhead between CPU and accelerators. The acceleration of our global atmospheric model is conducted to improve the use of the Intel MIC architecture. For the single-node test on Tianhe-2 against two Intel Ivy Bridge CPUs (24 cores), we can achieve 2.07×, 3.18×, and 4.35× speedups when using one, two, and three Intel Xeon Phi accelerators respectively. The average performance gain from SIMD vectorization on the Intel Xeon Phi processors is around 5× (out of the 8× theoretical case). Based on successful computation-communication overlapping, large-scale tests indicate that a nearly ideal weak-scaling efficiency of 93.5% is obtained when we gradually increase the number of nodes from 6 to 8,664 (nearly 1.7 million cores). In the strong-scaling test, the parallel efficiency is about 77% when the number of nodes increases from 1,536 to 8,664 for a fixed 65,664 × 5,664 × 6 mesh with 77.6 billion unknowns.

Developing highly scalable algorithms for global atmospheric modeling is becoming increasingly important as scientists inquire to understand behaviors of the global atmosphere at extreme scales. Nowadays, heterogeneous architecture based on both processors and accelerators is becoming an important solution for large-scale computing. However, large-scale simulation of the global atmosphere brings a severe challenge to the development of highly scalable algorithms that fit well into state-of-the-art heterogeneous systems. Although successes have been made on GPU-accelerated computing in some top-level applications, studies on fully exploiting heterogeneous architectures in global atmospheric modeling are still very less to be seen, due in large part to both the computational difficulties of the mathematical models and the requirement of high accuracy for long term simulations.