SDSL-Solver：可扩展的分布式稀疏线性求解器用于大规模内点法
SDSL-Solver: Scalable Distributed Sparse Linear Solvers for Large-Scale Interior Point Methods
AI总结：
本文提出SDSL-Solver，一种用于大规模内点法的可扩展分布式稀疏线性求解器，采用Krylov子空间方法和稀疏过滤技术，针对不同问题特性提供Block Jacobi和Bordered Block Diagonal两种并行方法，实验显示其在多节点集群上显著提升求解效率。
摘要：
稀疏线性系统的求解是内点法（IPMs）中的主要计算瓶颈，通常消耗总求解时间的70%以上。随着优化问题规模扩展到数百万变量，直接求解器面临不可接受的填充、过度的内存消耗和有限的并行扩展性。本文提出SDSL-Solver，一种针对IPMs的可扩展分布式稀疏线性求解器框架。SDSL-Solver采用Krylov子空间方法，结合基于数值的稀疏过滤和对角修正技术，生成高质量的预处理器。为适应不同的问题特性，SDSL-Solver提供两种互补的分布式并行方法：Block Jacobi用于对角占优矩阵，Bordered Block Diagonal（BBD）用于一般或病态矩阵，通过Schur补技术实现全局耦合的预处理。预处理器重用策略进一步摊销了连续IPMs迭代中的构造成本。我们在多节点集群上评估了SDSL-Solver，矩阵规模从数万到数百万不等，使用X86处理器。实验结果表明，在Block Jacobi和BBD分布式方法下，四节点配置的SDSL-Solver相比在同一节点数上的PETSc平均加速6.23倍，相比单节点PARDISO平均加速97.54倍和5.85倍。
英文摘要：
The solution of sparse linear systems constitutes the dominant computational bottleneck in interior point methods (IPMs), frequently consuming over 70% of the total solution time. As optimization problems scale to millions of variables, direct solvers encounter prohibitive fill-in, excessive memory consumption, and limited parallel scalability. We present SDSL-Solver, a scalable distributed sparse linear solver framework designed for IPMs. SDSL-Solver employs Krylov subspace methods, combined with numerics-based sparse filtering and diagonal correction techniques that produce high-quality preconditioners. To accommodate diverse problem characteristics, SDSL-Solver offers two complementary distributed parallel methods: Block Jacobi for diagonally dominant matrices, and Bordered Block Diagonal (BBD) for general or ill-conditioned matrices requiring globally coupled preconditioning via Schur complement techniques. A preconditioner reuse strategy further amortizes construction costs across consecutive IPMs iterations. We evaluate SDSL-Solver on benchmark problems with matrix dimensions ranging from tens of thousands to over five million on multi-node clusters equipped with X86 processors. The experimental results show that under the Block Jacobi and BBD distributed methods, SDSL-Solver on a four-node configuration achieves average speedups of 6.23 times and 7.77 times, respectively, compared to PETSc running on the same number of nodes. Relative to the single-node PARDISO, the average speedups reach 97.54 times and 5.85 times, respectively.
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