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An effective power based parallel preconditioner is proposed for generallarge sparse linear systems. The preconditioner combines a power seriesexpansion method with some low-rank correction techniques, where theSherman-Morrison-Woodbury formula is utilized. A matrix splitting of the Schurcomplement is proposed to expand the power series. The number of terms used inthe power series expansion can control the approximation accuracy of thepreconditioner to the inverse of the Schur complement. To construct thepreconditioner, graph partitioning is invoked to reorder the originalcoefficient matrix, leading to a special block two-by-two matrix whose twooff-diagonal submatrices are block diagonal. Variables corresponding tointerface variables are obtained by solving a linear system with the coeffcientmatrix being the Schur complement. For the variables related to the interiorvariables, one only needs to solve a block diagonal linear system. This can beperformed efficiently in parallel. Various numerical examples are provided toillustrate that the efficiency of the proposed preconditioner.

A power Schur complement Low-Rank correction preconditioner for general sparse linear systems