The solution of eigenproblems is often a key computational bottleneck that limits the tractable system size of numerical algorithms, among them electronic structure theory in chemistry and in condensed matter physics. Large eigenproblems can easily exceed the capacity of a single compute node, thus must be solved on distributed-memory parallel computers. We here present GPU-oriented optimizations of the ELPA two-stage tridiagonalization eigensolver (ELPA2). On top of its existing cuBLAS-based GPU offloading, we add a CUDA kernel to speed up the back-transformation of eigenvectors, which can be the computationally most expensive part of the two-stage tridiagonalization algorithm. We benchmark the performance of this GPU-accelerated eigensolver on two hybrid CPU-GPU architectures, namely a compute cluster based on Intel Xeon Gold CPUs and NVIDIA Volta GPUs, and the Summit supercomputer based on IBM POWER9 CPUs and NVIDIA Volta GPUs. Consistent with previous benchmarks on CPU-only architectures, the GPU-accelerated two-stage solver exhibits a parallel performance superior to the one-stage counterpart. Finally, we demonstrate the performance of the GPU-accelerated eigensolver developed in this work for routine semi-local KS-DFT calculations comprising thousands of atoms.

中文翻译：

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用于密集对称和厄米特特征问题的 ELPA2 分布式特征求解器的 GPU 加速

特征问题的解决通常是一个关键的计算瓶颈，它限制了数值算法的易处理系统规模，其中包括化学和凝聚态物理中的电子结构理论。大型特征问题很容易超过单个计算节点的容量，因此必须在分布式内存并行计算机上解决。我们在这里展示了 ELPA 两阶段三对角化特征求解器 (ELPA2) 的面向 GPU 的优化。在其现有的基于 cuBLAS 的 GPU 卸载之上，我们添加了一个 CUDA 内核来加速特征向量的反向转换，这可能是两阶段三对角化算法中计算成本最高的部分。我们在两个混合 CPU-GPU 架构上对这个 GPU 加速的特征求解器的性能进行了基准测试，即基于 Intel Xeon Gold CPU 和 NVIDIA Volta GPU 的计算集群，以及基于 IBM POWER9 CPU 和 NVIDIA Volta GPU 的 Summit 超级计算机。与之前在纯 CPU 架构上的基准测试一致，GPU 加速的两级求解器表现出优于单级求解器的并行性能。最后，我们展示了在这项工作中开发的 GPU 加速特征求解器的性能，用于包含数千个原子的常规半局部 KS-DFT 计算。