parGeMSLR：一般稀疏矩阵的并行多级Schur补低秩预处理及求解包
parGeMSLR: A Parallel Multilevel Schur Complement Low-Rank Preconditioning and Solution Package for General Sparse Matrices
摘要：
本文讨论了parGeMSLR，一个C++/MPI软件库，用于在分布式存储计算环境下通过预条件Krylov子空间方法求解稀疏线性代数方程组。parGeMSLR中实现的预条件子是基于代数区域分解和分区的对称邻接图递归到几个不重叠的分区通过一个p路顶点分离器，其中p是整数倍的MPI进程的总数。从数值的角度来看，parGeMSLR建立了一个舒尔补近似逆预条件子之间的总和界面耦合矩阵的矩阵逆和低秩校正项。为了减少与近似逆矩阵的计算相关联的成本，parGeMSLR利用代数域的多级划分。parGeMSLR库是在消息传递接口之上实现的，可以求解实数和复杂的线性系统。此外，parGeMSLR可以利用具有对一个或多个图形处理单元的节点内访问的混合计算环境。最后，parGeMSLR的并行效率（弱和强标度）上所产生的三维偏微分方程的离散化的几个模型问题证明。
英文摘要：
This paper discusses parGeMSLR, a C++/MPI software library for the solution of sparse systems of linear algebraic equations via preconditioned Krylov subspace methods in distributed-memory computing environments. The preconditioner implemented in parGeMSLR is based on algebraic domain decomposition and partitions the symmetrized adjacency graph recursively into several non-overlapping partitions via a p-way vertex separator, where p is an integer multiple of the total number of MPI processes. From a numerical perspective, parGeMSLR builds a Schur complement approximate inverse preconditioner as the sum between the matrix inverse of the interface coupling matrix and a low-rank correction term. To reduce the cost associated with the computation of the approximate inverse matrices, parGeMSLR exploits a multilevel partitioning of the algebraic domain. The parGeMSLR library is implemented on top of the Message Passing Interface and can solve both real and complex linear systems. Furthermore, parGeMSLR can take advantage of hybrid computing environments with in-node access to one or more Graphics Processing Units. Finally, the parallel efficiency (weak and strong scaling) of parGeMSLR is demonstrated on a few model problems arising from discretizations of 3D Partial Differential Equations.
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