Parallel Conjugate Gradient Preconditioning via Incomplete Cholesky of Overlapping Submatrices

Abstract

A coarse-grain parallel iterative algorithm for the numerical solution of large sparse symmetric positive definite (SPD) linear systems is described. Based on the new convergence theory for the Preconditioned Conjugate Gradient (PCG) method, an overlapping domain decomposition-like preconditioning is developed. The preconditioning techniques combine the Block Incomplete Inverse Cholesky with approximate inversion of sub matrices via the 2nd order Incomplete Cholesky factorization. Numerical results obtained with an MPI-based FORTRAN code demonstrate good parallel performance on a set of large-scale ill-conditioned problems. It shows that harder the problem is, greater is the gain in performance of the proposed method as compared to other commonly used parallel iterative solvers. The parallel properties of the solver appear to be as good as expected from our theoretical considerations.