MPI+OpenMP implementation of the conjugate gradient method with IC(0) preconditioner based on the use of mesh node reordering

Abstract

The paper proposes methods for using MPI and MPI+OpenMP technologies to construct and invert 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. 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. The use of OpenMP technology in the construction and inversion of the preconditioner is carried out for most rows of the matrix. Comparative timing results for the MPI+OpenMP and MPI implementations of the proposed preconditioning used with the conjugate gradient method for a model problems and the sparse matrix collections SuiteSparse are presented.