Low Rank Off-diagonal Block Preconditioners for Solving Sparse Linear Systems on Parallel Computers

For a sparse linear system Ax = b, preconditioners of the form C = D + L+ U , where D is the block diagonal part of A (or incomplete factorization approximation of its blocks), and L and U are block strictly lower and upper triangular matrices composed of low-ranks approximations of the respective blocks of A, are examined. C is applied directly, by solving Cz = w, or partially, by applying one step of BSSOR to Cz = w. Use of low-rank approximations of o -diagonal blocks is common in dense systems, but apparently has not been considered for sparse systems. This paper examines ways of de ning the o -diagonal blocks and provides a detailed analysis for systems occuring in solving Laplace equation on a uniform mesh. Methods of applying C as a parallel preconditioner are proposed and analyzed, their cost being compared to that of applying a Jacobi preconditioner D. Testing results are presented, comparing the use of low-rank approximations with Jacobi and block SSOR preconditioning.