The symmetric sparse matrix-vector multiplication (SymmSpMV) is an important building block for many numerical linear algebra kernel operations or graph traversal applications. Parallelizing SymmSpMV on today's multicore platforms with up to 100 cores is difficult due to the need to manage conflicting updates on the result vector. Coloring approaches can be used to solve this problem without data duplication, but existing coloring algorithms do not take load balancing and deep memory hierarchies into account, hampering scalability and full-chip performance. In this work, we propose the recursive algebraic coloring engine (RACE), a novel coloring algorithm and open-source library implementation that eliminates the shortcomings of previous coloring methods in terms of hardware efficiency and parallelization overhead. We describe the level construction, distance-k coloring, and load balancing steps in RACE, use it to parallelize SymmSpMV, and compare its performance on 31 sparse matrices with other state-of-the-art coloring techniques and Intel MKL on two modern multicore processors. RACE outperforms all other approaches substantially. By means The project is funded by the German DFG priority programme 1648 "Software for Exascale Computing (SPPEXA)" and the Swiss National Science Foundation (SNF) under the projects "Dual-Phase Steels-From Micro to Macro Properties (EXASTEEL-2)" (DFG, SNF) and "Equipping Sparse Solvers for Exascale (ESSEX-II)" (DFG).

Hardware platforms in high performance computing are constantly getting more complex to handle even when considering multicore CPUs alone. Numerous features and configuration options in the hardware and the software environment that are relevant for performance are not even known to most application users or developers. Microbenchmarks, i.e., simple codes that fathom a particular aspect of the hardware, can help to shed light on such issues, but only if they are well understood and if the results can be reconciled with known facts or performance models. The insight gained from microbenchmarks may then be applied to real applications for performance analysis or optimization. In this paper we investigate two modern Intel x86 server CPU architectures in depth: Broadwell EP and Cascade Lake SP. We highlight relevant hardware configuration settings that can have a decisive impact on code performance and show how to properly measure on-chip and off-chip data transfer bandwidths. The new victim L3 cache of Cascade Lake and its advanced replacement policy receive due attention. Finally we use DGEMM, sparse matrix-vector multiplication, and the HPCG benchmark to make a connection to relevant application scenarios.

No abstract

The A64FX CPU is arguably the most powerful Arm-based processor design to date.Although it is a traditional cache-based multicore processor, its peak performance and memory bandwidth rival accelerator devices. A good understanding of its performance features is of paramount importance for developers who wish to leverage its full potential. We present an architectural analysis of the A64FX used in the Fujitsu FX1000 supercomputer at a level of detail that allows for the construction of Execution-Cache-Memory performance models for steady-state loops. In the process we identify architectural peculiarities that point to viable generic optimization strategies. After validating the model using simple streaming loops we apply the insight gained to sparse matrix-vector multiplication (SpMV) and the domain wall (DW) kernel from quantum chromodynamics. For SpMV we show why the compressed row storage (CRS) matrix storage format is not a good practical choice on this architecture and how the SELL-C-𝜎 format can achieve bandwidth saturation. For the DW kernel we provide a cache-reuse analysis and show how an appropriate choice of data layout for complex arrays can realize memory-bandwidth saturation in this case as well. A comparison with state-of-the-art high-end Intel Cascade Lake AP and Nvidia V100 systems puts the capabilities of the A64FX into perspective. We also explore the potential for power optimizations using the tuning knobs provided by the Fugaku system, achieving energy savings of about 31% for SpMV and 18% for DW.

No abstract