大规模特征值计算中的正交并行层次
Orthogonal layers of parallelism in large-scale eigenvalue computations
AI总结：
针对大规模特征值求解中分布式稀疏矩阵向量乘法的通信开销问题，提出通过水平与垂直两个正交并行层次及不同分布式向量布局来恢复可扩展性，理论分析与基准测试验证了其有效性。
摘要：
我们以 filter diagonalization 为例，研究大规模 eigensolvers 中分布式稀疏矩阵-（多）向量乘法的通信开销问题。研究基础是一个性能模型，包含一个无需运行代码即可直接从矩阵稀疏模式计算的通信度量。该模型量化了因通信开销导致可扩展性和并行效率受损的程度。为恢复可扩展性，我们在 filter diagonalization 技术中识别出两个正交并行层次：水平层将稀疏矩阵的行分布到各进程；垂直层将多向量束分布到不同进程组。基于通信度量的分析预测，当且仅当通过不同的分布式向量布局实现这两个正交并行层次时，才能恢复可扩展性。我们的理论分析得到了来自量子与固体物理、道路网络及非线性规划的应用矩阵基准测试的验证。最后，我们通过两个示例应用案例——一个 exciton 和一个强相关电子系统——展示了使用正交并行层次的益处，它们分别产生较小或较大的通信开销。
英文摘要：
We address the communication overhead of distributed sparse matrix-(multiple)-vector multiplication in the context of large-scale eigensolvers, using filter diagonalization as an example. The basis of our study is a performance model which includes a communication metric that is computed directly from the matrix sparsity pattern without running any code. The performance model quantifies to which extent scalability and parallel efficiency are lost due to communication overhead. To restore scalability, we identify two orthogonal layers of parallelism in the filter diagonalization technique. In the horizontal layer the rows of the sparse matrix are distributed across individual processes. In the vertical layer bundles of multiple vectors are distributed across separate process groups. An analysis in terms of the communication metric predicts that scalability can be restored if, and only if, one implements the two orthogonal layers of parallelism via different distributed vector layouts. Our theoretical analysis is corroborated by benchmarks for application matrices from quantum and solid state physics, road networks, and nonlinear programming. We finally demonstrate the benefits of using orthogonal layers of parallelism with two exemplary application cases -- an exciton and a strongly correlated electron system -- which incur either small or large communication overhead.
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