2020/02/03 by Qingqing Zheng, Zheng, Qingqing, Yuanzhe Xi +3 · 3 citations
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Applied mathematics #Block matrix #Coefficient matrix #Combinatorics #Diagonal #Eigenvalues and eigenvectors #Electromagnetic Scattering and Analysis #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Geometry #Iterative method #Linear system #Mathematical analysis #Mathematical optimization #Mathematics #Matrix (chemical analysis) #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Power iteration #Power series #Preconditioner #Rank (graph theory) #Schur complement #Sparse matrix #cs.NA #math.NA
paper · pdf · doi:10.48550/arxiv.2002.00917
published in arXiv (Cornell University) (Cornell University)
arxiv created 2020/02/03 · openalex publication_date 2020/02/03 · arxiv updated 2020/02/04 · openalex created_date 2020/02/07 · openalex updated_date 2026/08/06
An effective power based parallel preconditioner is proposed for general large sparse linear systems. The preconditioner combines a power series expansion method with some low-rank correction techniques, where the Sherman-Morrison-Woodbury formula is utilized. A matrix splitting of the Schur complement is proposed to expand the power series. The number of terms used in the power series expansion can control the approximation accuracy of the preconditioner to the inverse of the Schur complement. To construct the preconditioner, graph partitioning is invoked to reorder the original coefficient matrix, leading to a special block two-by-two matrix whose two off-diagonal submatrices are block diagonal. Variables corresponding to interface variables are obtained by solving a linear system with the coeffcient matrix being the Schur complement. For the variables related to the interior variables, one only needs to solve a block diagonal linear system. This can be performed efficiently in parallel. Various numerical examples are provided to illustrate that the efficiency of the proposed preconditioner.