Some Ways of Parallel Implementation of the Conjugate Gradient Method with an Implicit Factorized Preconditioner

Abstract

The paper considers two ways of using MPI and MPI + OpenMP technologies for constructing and inverting the preconditioner of an incomplete triangular Cholesky decomposition without filling IC(0) for solving systems of linear algebraic equations with an arbitrary symmetric positive definite matrix. They differ in the way in which the preconditioning matrix IC(0) is computed. Methods of using MPI and MPI + OpenMP technologies are based on the use of grid node orderings consistent with the division of the calculation area. Comparative timing results for the MPI + OpenMP and MPI implementations of the proposed preconditioning IC(0) used with the conjugate gradient method for a model problems and the sparse matrix collections SuiteSparse, as well as comparing the time required to solve these problems using the two MPI and MPI + OpenMP technology methods, are presented.