SIAM Journal on Scientific Computing, Ahead of Print.
Linear solvers for large and sparse systems are a key element of scientific applications, and their efficient implementation is necessary to harness the computational power of current computers. Algebraic MultiGrid (AMG) preconditioners are a popular ingredient of such linear solvers; this is the motivation for the present work, where we examine some recent developments in a package of AMG preconditioners to improve efficiency, scalability, and robustness on extreme scale problems. The main novelty is the design and implementation of a parallel coarsening algorithm based on aggregation of unknowns employing weighted graph matching techniques; this is a completely automated procedure, requiring no information from the user, and applicable to general symmetric positive definite (s.p.d.) matrices. The new coarsening algorithm improves in terms of numerical scalability at low operator complexity upon decoupled aggregation algorithms available in previous releases of the package. The preconditioners package is built on the parallel software framework \tt PSBLAS, which has also been updated to progress towards exascale. We present weak scalability results on one of the most powerful supercomputers in Europe for linear systems with sizes up to $O(10^{10})$ unknowns.

中文翻译：

---

面向极端规模的线性求解器的 AMG 预处理器

SIAM 科学计算杂志，提前印刷。
用于大型和稀疏系统的线性求解器是科学应用的关键要素，它们的有效实施对于利用当前计算机的计算能力是必要的。代数多重网格 (AMG) 预处理器是此类线性求解器的流行成分；这就是目前工作的动机，我们研究了 AMG 预处理器包的一些最新进展，以提高极端规模问题的效率、可扩展性和鲁棒性。主要新颖之处是基于未知数聚合的并行粗化算法的设计和实现，采用加权图匹配技术；这是一个完全自动化的过程，不需要用户提供信息，适用于一般对称正定 (spd) 矩阵。新的粗化算法在低算子复杂度的数值可扩展性方面改进了先前版本包中可用的解耦聚合算法。预处理器包建立在并行软件框架 \tt PSBLAS 上，该框架也已更新以向百亿亿级发展。我们在欧洲最强大的超级计算机之一上为线性系统提供了弱可扩展性结果，未知大小高达 $O(10^{10})$。