In \emph{Wang et al., A Shifted Laplace Rational Filter for Large-Scale Eigenvalue Problems}, the SLRF method was proposed to compute all eigenvalues of a symmetric definite generalized eigenvalue problem lying in an interval on the real positive axis. The current paper discusses a parallel implementation of this method, abbreviated as ParaSLRF. The parallelization consists of two levels: (1) on the highest level, the application of the rational filter to the various vectors is partitioned among groups of processors; (2) within each group, every linear system is solved in parallel. In ParaSLRF, the linear systems are solved by iterative methods instead of direct ones, in contrast to other rational filter methods, such as, PFEAST. Because of the specific selection of poles in ParaSLRF, the computational cost of solving the associated linear systems for each pole, is almost the same. This intrinsically leads to a better load balance between each group of resources, and reduces waiting times of processes. We show numerical experiments from finite element models of mechanical vibrations, and show a detailed parallel performance analysis. ParaSLRF shows the best parallel efficiency, compared to other rational filter methods based on quadrature rules for contour integration. To further improve performance, the converged eigenpairs are locked, and a good initial guess of iterative linear solver is proposed. These enhancements of ParaSLRF show good two-level strong scalability and excellent load balance in our experiments.

翻译：在《Wang等人，一种用于大规模特征值问题的平移拉普拉斯有理滤波器》中，提出了SLRF方法来计算位于实正轴上某一区间内的对称正定广义特征值问题的所有特征值。本文讨论了该方法的并行实现，简称为ParaSLRF。并行化包含两个层次：(1) 在最高层，有理滤波器对不同向量的应用被分配到处理器组之间；(2) 在每个组内，每个线性系统被并行求解。在ParaSLRF中，线性系统通过迭代法而非直接法求解，这与其他有理滤波器方法（例如PFEAST）形成对比。由于ParaSLRF中对极点的特定选择，求解每个极点对应线性系统的计算成本几乎相同。这本质上带来了各资源组之间更好的负载平衡，并减少了进程的等待时间。我们展示了来自机械振动有限元模型的数值实验，并提供了详细的并行性能分析。与基于围道积分求积法则的其他有理滤波器方法相比，ParaSLRF展现出最佳的并行效率。为了进一步提升性能，收敛的特征对会被锁定，并且提出了迭代线性求解器的良好初始猜测。这些对ParaSLRF的增强在我们的实验中展现了良好的两级强可扩展性和优异的负载平衡。